Nanophotonics 9781847045782, 9781905209286, 1905209282

Citation preview

Nanophotonics

Nanophotonics

Edited by Hervé Rigneault Jean-Michel Lourtioz Claude Delalande Ariel Levenson

First published in France in 2005 by Hermes Science/Lavoisier entitled “La nanophotonique” First published in Great Britain and the United States in 2006 by ISTE Ltd Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 6 Fitzroy Square London W1T 5DX UK

ISTE USA 4308 Patrice Road Newport Beach, CA 92663 USA

www.iste.co.uk © ISTE Ltd, 2006 © GET and LAVOISIER, 2005 The rights of Hervé Rigneault, Jean-Michel Lourtioz, Claude Delalande and Ariel Levenson to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Nanophotonique. English. Nanophotonics / Hervé Rigneault ... [et al.]. p. cm. Includes index. ISBN-13: 978-1-905209-28-6 ISBN-10: 1-905209-28-2 1. Photonics. 2. Nanotechnology. I. Rigneault, Hervé. II. Title. TA1520.N35 2006 621.36--dc22 2006008801 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 10: 1-905209-28-2 ISBN 13: 978-1-905209-28-6 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire.

Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Chapter 1. Photonic Crystals: From Microphotonics to Nanophotonics . . . Pierre VIKTOROVITCH

17

1.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Reminders and prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1. Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1.1. Optical modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1.2. Dispersion characteristics . . . . . . . . . . . . . . . . . . . . . 1.2.2. A simple case: three-dimensional and homogeneous free space . 1.2.3. Structuration of free space and optical mode engineering . . . . . 1.2.4. Examples of space structuration: objects with reduced dimensionality 1.2.4.1. Two 3D sub-spaces . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4.2. Two-dimensional isotropic propagation: planar cavity. . . . 1.2.4.3. One-dimensional propagation: photonic wire . . . . . . . . . 1.2.4.4. Case of index guiding (two- or one-dimensionality) . . . . . 1.2.4.5. Zero-dimensionality: optical (micro)-cavity . . . . . . . . . . 1.2.5. Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. 1D photonic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1. Bloch modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2. Dispersion characteristics of a 1D periodic medium . . . . . . . . 1.3.2.1. Genesis and description of dispersion characteristics . . . . 1.3.2.2. Density of modes along the dispersion characteristics . . . . 1.3.3. Dynamics of Bloch modes. . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.1. Coupled mode theory . . . . . . . . . . . . . . . . . . . . . . . 1.3.3.2. Lifetime of a Bloch mode . . . . . . . . . . . . . . . . . . . . . 1.3.3.3. Merit factor of a Bloch mode. . . . . . . . . . . . . . . . . . . 1.3.4. The distinctive features of photonic crystals . . . . . . . . . . . . .

17 19 19 20 20 20 21 22 22 24 25 26 26 27 28 29 30 30 32 33 33 34 35 35

6

Nanophotonics

1.3.5. Localized defect in a photonic band gap or optical microcavity 1.3.5.1. Donor and acceptor levels. . . . . . . . . . . . . . . . . . . . 1.3.5.2. Properties of cavity modes in a 1DPC . . . . . . . . . . . . 1.3.5.3. Fabry-Pérot type optical filter . . . . . . . . . . . . . . . . . 1.3.6. 1D photonic crystal in a dielectric waveguide and waveguided Bloch modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6.1. Various diffractive coupling processes between optical modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6.2. Determination of the dispersion characteristics of waveguided Bloch modes . . . . . . . . . . . . . . . . . . . . . . . 1.3.6.3. Lifetime and merit factor of waveguided Bloch modes: radiation optical losses . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6.4. Localized defect or optical microcavity . . . . . . . . . . . 1.3.7. Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. 3D photonic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1. From dream … . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2. … to reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. 2D photonic crystals: the basics . . . . . . . . . . . . . . . . . . . . . . . 1.5.1. Conceptual tools: Bloch modes, direct and reciprocal lattices, dispersion curves and surfaces . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1.1. Bloch modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1.2. Direct and reciprocal lattices . . . . . . . . . . . . . . . . . . 1.5.1.3. Dispersion curves and surfaces. . . . . . . . . . . . . . . . . 1.5.2. 2D photonic crystal in a planar dielectric waveguide . . . . . . . 1.5.2.1. An example of the potential of 2DPC in terms of angular resolution: the super-prism effect . . . . . . . . . . . . . . . . . . . . 1.5.2.2. Strategies for vertical confinement in 2DPC waveguided configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. 2D photonic crystals: basic building blocks for planar integrated photonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1. Fabrication: a planar technological approach. . . . . . . . . . . . 1.6.1.1. 2DPC formed in an InP membrane suspended in air . . . . 1.6.1.2. 2DPC formed in an InP membrane bonded onto silica on silicon by molecular bonding . . . . . . . . . . . . . . 1.6.2. Localized defect in the PBG or microcavity . . . . . . . . . . . . 1.6.3. Waveguiding structures . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3.1. Propagation losses in a straight waveguide . . . . . . . . . 1.6.3.2. Bends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3.3. The future of PC-based waveguides lies principally in the guiding of light . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4. Wavelength selective transfer between two waveguides . . . . . 1.6.5. Micro-lazers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

36 37 38 39

.

40

.

40

.

42

. . . . . . .

43 44 46 46 46 47 49

. . . . .

50 50 51 52 54

.

56

.

57

. . .

59 59 59

. . . . .

60 62 64 66 67

. . .

69 70 73

Table of Contents

1.6.5.1. Threshold power . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.5.2. Example: the case of the surface emitting Bloch mode lazer 1.6.6. Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. Towards 2.5-dimensional Microphotonics . . . . . . . . . . . . . . . . . 1.7.1. Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8. General conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2. Bidimensional Photonic Crystals for Photonic Integrated Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anne TALNEAU 2.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The three dimensions in space: planar waveguide perforated by a photonic crystal on InP substrate . . . . . . . . . . . . . . . . . . 2.2.1. Vertical confinement: a planar waveguide on substrate . 2.2.2. In-plane confinement: intentional defects within the gap. 2.2.2.1. Localized defects . . . . . . . . . . . . . . . . . . . . . 2.2.2.2. Linear defects . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Technology for drilling holes on InP-based materials . . . . . . 2.3.1. Mask generation. . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Dry-etching of InP-based semiconductor materials . . . . 2.4. Modal behavior and performance of structures . . . . . . . . . . 2.4.1. Passive structures . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1.1. Straight guides, taper . . . . . . . . . . . . . . . . . . 2.4.1.2. Bend, combiner . . . . . . . . . . . . . . . . . . . . . . 2.4.1.3. Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Active structures: lazers . . . . . . . . . . . . . . . . . . . . 2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

109

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . . . . . . . . . . . . .

85

Chapter 3. Photonic Crystal Fibers . . . . . . . . . . . . . . . . . . . . . . . . . Dominique PAGNOUX . . . . . .

. . . . . . . . . . . . . . . . .

85

86 86 87 88 88 89 90 90 91 92 92 93 96 100 102 104 105

. . . . . .

. . . . . . . . . . . . . . . . .

74 75 77 77 77 80 81 82

. . . . . . . . . . . . . . . . .

3.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . 3.2. Two guiding principles in microstructured fibers 3.3. Manufacture of microstructured fibers . . . . . . . 3.4. Modeling TIR-MOFs . . . . . . . . . . . . . . . . . 3.4.1. The “effective-V model”. . . . . . . . . . . . 3.4.2. Modal methods for calculating the fields . .

. . . . . . . . . . . . . . . . .

7

. . . . . .

. . . . . .

109 112 116 117 117 118

8

Nanophotonics

3.5. Main properties and applications of TIR-MOFs . . 3.5.1. Single mode propagation . . . . . . . . . . . . 3.5.2. Propagation loss . . . . . . . . . . . . . . . . . . 3.5.3. Chromatic dispersion . . . . . . . . . . . . . . . 3.5.4. Birefringence . . . . . . . . . . . . . . . . . . . 3.5.5. Non-conventional effective areas. . . . . . . . 3.6. Photonic bandgap fibers . . . . . . . . . . . . . . . . 3.6.1. Propagation in photonic bandgap fibers . . . . 3.6.2. Some applications of photonic crystal fibers . 3.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 3.8. References . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

120 120 120 121 123 124 125 125 127 128 129

Chapter 4. Quantum Dots in Optical Microcavities . . . . . . . . . . . . . . . Jean-Michel GÉRARD

135

4.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Building blocks for solid-state CQED . . . . . . . . . . . . . . 4.2.1. Self-assembled QDs as “artificial atoms” . . . . . . . . . 4.2.2. Solid-state optical microcavities . . . . . . . . . . . . . . 4.3. QDs in microcavities: some basic CQED experiments . . . . 4.3.1. Strong coupling regime . . . . . . . . . . . . . . . . . . . 4.3.2. Weak coupling regime: enhancement/inhibition of the SE rate and “nearly” single mode SE . . . . . . . . . . . 4.3.3. Applications of CQED effects to single photon sources and nanolazers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

145

. . . . . . . . . . . .

150 154

Chapter 5. Nonlinear Optics in Nano- and Microstructures. . . . . . . . . . Yannick DUMEIGE and Fabrice RAINERI

159

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . .

135 137 137 139 142 142

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . .

. . . . . .

5.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Introduction to nonlinear optics . . . . . . . . . . . . . 5.2.1. Maxwell equations and nonlinear optics . . . . 5.2.2. Second order nonlinear processes . . . . . . . . 5.2.2.1. Three wave mixing. . . . . . . . . . . . . . 5.2.2.2. Second harmonic generation . . . . . . . . 5.2.2.3. Parametric amplification . . . . . . . . . . 5.2.2.4. How can phase matching be achieved? . . 5.2.2.5. Applications of second order nonlinearity 5.2.3. Third order processes. . . . . . . . . . . . . . . . 5.2.3.1. Four wave mixing . . . . . . . . . . . . . . 5.2.3.2. Optical Kerr effect . . . . . . . . . . . . . .

. . . . . .

. . . . . . . . . . .

. . . . . . . . . . . .

. . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

159 160 160 164 165 166 169 170 173 173 173 175

Table of Contents

5.2.3.3. Nonlinear spectroscopy: Raman, Brillouin and Rayleigh scatterings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Nonlinear optics of nano- or microstructured media. . . . . . . . . . 5.3.1. Second order nonlinear optics in III–V semiconductors . . . . 5.3.1.1. Quasi-phase matching in III–V semiconductors. . . . . . 5.3.1.2. Quasi-phase matching in microcavity. . . . . . . . . . . . 5.3.1.3. Bidimensional quasi-phase matching . . . . . . . . . . . . 5.3.1.4. Form birefringence . . . . . . . . . . . . . . . . . . . . . . . 5.3.1.5. Phase matching in one-dimensional photonic crystals . . 5.3.1.6. Phase matching in two-dimensional photonic crystal waveguide . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Third order nonlinear effects . . . . . . . . . . . . . . . . . . . . 5.3.2.1. Continuum generation in microstructured optical fibers . 5.3.2.2. Optical reconfiguration of two-dimensional photonic crystal slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2.3. Spatial solitons in microcavities . . . . . . . . . . . . . . . 5.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

177 177 178 178 179 180 180 181

. . . . . .

183 184 184

. . . .

. . . .

184 186 187 187

Chapter 6. Third Order Optical Nonlinearities in Photonic Crystals . . . . Robert FREY, Philippe DELAYE and Gérald ROOSEN

191

6.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . 6.2. Third order nonlinear optic reminder. . . . . . . . . 6.2.1. Third order optical nonlinearities. . . . . . . . 6.2.2. Some third order nonlinear optical processes. 6.2.3. Influence of the local field. . . . . . . . . . . . 6.3. Local field in photonic crystals . . . . . . . . . . . . 6.4. Nonlinearities in photonic crystals . . . . . . . . . . 6.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 6.6. References . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

191 192 192 194 196 198 203 204 204

Chapter 7. Controling the Optical Near Field: Implications for Nanotechnology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frédérique DE FORNEL

207

7.1. Introduction. . . . . . . . . . . . . . . . . . . . . . 7.2. How is the near field defined? . . . . . . . . . . . 7.2.1. Dipolar emission . . . . . . . . . . . . . . . 7.2.2. Diffraction by a sub-wavelength aperture. 7.2.3. Total internal reflection . . . . . . . . . . . 7.3. Optical near field microscopies . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . . . . .

. . . . . .

. . . . . . . . .

. . . . . .

. . . . . . . . .

. . . . . .

. . . . . . . . .

. . . . . .

. . . . . . . . .

. . . . . .

. . . . . . . . .

. . . . . .

. . . . . . . . .

. . . . . .

. . . . . . . . .

. . . . . .

. . . . . . . . .

. . . . . .

. . . . . . . . .

. . . . . . . .

9

. . . . . .

. . . . . . . . .

. . . . . .

. . . . . .

207 208 208 212 213 217

10

Nanophotonics

7.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. Fundamental principles . . . . . . . . . . . . . . . . . . . . 7.3.3. Realization of near field probes. . . . . . . . . . . . . . . . 7.3.4. Imaging methods in near field optical microscopes . . . . 7.3.5. Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.6. What is actually measured in near field? . . . . . . . . . . 7.3.7. PSTM configuration . . . . . . . . . . . . . . . . . . . . . . 7.3.8. Apertureless microscope . . . . . . . . . . . . . . . . . . . . 7.3.9. Effect of coherence on the structure of near field images 7.4. Characterization of integrated-optical components . . . . . . . 7.4.1. Characterization of guided modes . . . . . . . . . . . . . . 7.4.2. Photonic crystal waveguides . . . . . . . . . . . . . . . . . 7.4.3. Excitation of cavity modes . . . . . . . . . . . . . . . . . . 7.4.4. Localized generation of surface plasmons . . . . . . . . . 7.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

217 217 219 220 222 223 223 225 226 227 227 229 230 232 235 236

Chapter 8. Sub-Wavelength Optics: Towards Plasmonics . . . . . . . . . . Alain DEREUX

239

8.1. Technological context. . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Detecting optical fields at the sub-wavelength scale. . . . . . . . . 8.2.1. Principle of sub-wavelength measurement . . . . . . . . . . . 8.2.2. Scattering theory of electromagnetic waves . . . . . . . . . . 8.2.3. Electromagnetic LDOS . . . . . . . . . . . . . . . . . . . . . . 8.2.4. PSTM detection of the electric or magnetic components of optical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5. SNOM detection of the electromagnetic LDOS . . . . . . . . 8.3. Localized plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1. Squeezing of the near-field by localized plasmons coupling. 8.3.2. Controling the coupling of localized plasmons. . . . . . . . . 8.4. Sub–λ optical devices . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1. Coupling in. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2. Sub–λ waveguides . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3. Towards plasmonics: plasmons on metal stripes. . . . . . . . 8.4.4. Prototypes of submicron optical devices . . . . . . . . . . . . 8.5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

239 240 240 242 244

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

246 247 249 250 251 254 254 254 255 256 263

Table of Contents

Chapter 9. The Confined Universe of Electrons in Semiconductor Nanocrystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . Maria CHAMARRO 9.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Electronic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1. “Naif” model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1.1. Absorption and luminescence spectra. . . . . . . . . . . . . 9.2.2. Fine electronic structure . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2.1. Size-selective excitation. . . . . . . . . . . . . . . . . . . . . 9.2.2.2. “Dark” electron-hole pair . . . . . . . . . . . . . . . . . . . . 9.3. Micro-luminescence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Auger effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5. Applications in nanophotonics . . . . . . . . . . . . . . . . . . . . . . . 9.5.1. Semiconductor nanocrystals: single photon sources . . . . . . . 9.5.2. Semiconductor nanocrystals: new fluorescent labels for biology . 9.5.3. Semiconductor nanocrystals: a new active material for tunable lazers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

265

. . . . . . . . . . . .

265 266 266 269 271 271 274 276 279 281 281 283

. . .

285 286 287

Chapter 10. Nano-Biophotonics . . . . . . . . . . . . . . . . . . . . . . . . . . . Hervé RIGNEAULT and Pierre-François LENNE

293

10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. The cell: scale and constituents . . . . . . . . . . . . . . . . . . 10.3. Origin and optical contrast mechanisms . . . . . . . . . . . . . 10.3.1. Classical contrast mechanisms: bright field, dark field, phase contrast and interferometric contrast. . . . . . . . . . . . . 10.3.2. The fluorescence contrast mechanism . . . . . . . . . . . 10.3.2.1. The lifetime contrast . . . . . . . . . . . . . . . . . . 10.3.2.2. Resolving power in fluorescence microscopy . . . 10.3.3. Non-linear microscopy . . . . . . . . . . . . . . . . . . . . 10.3.3.1. Second harmonic generation (SHG) . . . . . . . . . 10.3.3.2. Coherent anti-Stokes Raman scattering (CARS) . 10.4. Reduction of the observation volume . . . . . . . . . . . . . . . 10.4.1. Far field methods . . . . . . . . . . . . . . . . . . . . . . . 10.4.1.1. 4Pi microscopy . . . . . . . . . . . . . . . . . . . . . 10.4.1.2. Microscopy on a mirror . . . . . . . . . . . . . . . . 10.4.1.3. Stimulated emission depletion: STED. . . . . . . .

. . . . . . . . . . . . . . .

293 295 296

. . . . . . . . . . . .

297 298 300 301 303 304 305 307 308 308 309 309

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

12

Nanophotonics

10.4.2. Near field methods 10.4.2.1. NSOM . . . . 10.4.2.2. TIRF . . . . . 10.4.2.3. Nanoholes . . 10.5. Conclusion . . . . . . . . 10.6. References. . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

311 312 312 313 314 314

List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

319

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

323

Preface

With the continuous miniaturization of electronic components over the last 50 years we have grown accustomed to the idea of micro-electronics where transistors are measured in microns, and today, with the advent of transistor grid lengths of around 10 nanometers, we are getting used to nano-electronics. Besides, we should not forget about Moore’s law1, a predictive law, according to which the length of the transistor grid is reduced by a factor of two approximately every 18 months. The concept of nanophotonics, although not surprising, remains, however, less clearly understood by the scientific community than that of nano-electronics. Admittedly, we realize that optoelectronic components, such as the lazer diode, modulators and detectors developed for the needs of optical telecommunications, are small nowadays, but there does not exist a Moore’s law of optoelectronics and the most usual limit naively imagined for optics is that of wavelength, i.e. a size close to the micron for waves of the visible and near infrared spectrum. It is, therefore, the main objective of this work to try and give a more precise overview of the rapidly emerging field of nanophotonics, wherein optical fields at the scale of a fraction of wavelength and even mainly sub-wavelength are sought to be controlled and designed. In fact, if the optical “chip” does not exist in the liking of the electronic “chip”, photonic crystals have recently led to great hopes for large-scale integration of optoelectronic components. Two-dimensional photonic crystals obtained through periodic structuring of a planar optical waveguide, in particular, have many characteristics which bring them closer to electronic micro- and nanostructures. In a simple vision, it suffices to introduce periodicity defects at suitably selected spots within the crystal to obtain the desired optical components (waveguides, bending light, micro-resonators, filters, etc.) and to pair them up with each other to form true 1 G. Moore, founder of INTEL.

14

Nanophotonics

photonic circuits. Admittedly, reality is more difficult than it appears, if only for the precision needed in the manufacturing of structures. In many cases it is considered lower or equal to 10 nanometers, and then all the relevance of parallels between nano-electronics and nanophotonics become apparent. The first two chapters are thus mainly dedicated to photonic crystals in planar optics, referring to other recently published works on the subject2, while focusing on the photonic components themselves, the dynamics of the photons plunged into a periodically structured medium and the prospect of obtaining high integration photonic circuits. On the subject of two-dimensional photonic crystals radically differing from planar guided optics, Chapter 3 tackles the topic of photonic crystals fibers and, more generally, of structured fibers. Not only is the propagation of light achieved then perpendicular to the plane of periodic structuring, but also the unique production technology is based on the first assembly performed on a macroscopic scale, the final micro-nano-structures obtained by a stretching process at the second stage. It is impressive to be able “to unravel” micro-nanophotonics over distances of several kilometers! From a practical standpoint, microstructured fibers and photonic crystal fibers open up unprecedented prospects with respect to the control of the propagation mode in fiber-optics and to the control of chromatic dispersion. By controlling optical confinement, we may also easily control the processes of nonlinear optics that can be developed within these fibers. Before the concepts of photonic circuit or fiber even appear, it should be remembered that the first studies of photonic crystals and structured materials for optics had been motivated, at the beginning of the 1980s, by the desire to control and even inhibit spontaneous emission in optoelectronic components. The largely conveyed emblematic image is that of the single transmitter in a uni-modal microcavity, every emitted photon being in the unique electromagnetic mode of the cavity. That aside, for the image to become reality over time, it was initially necessary to control the realization of nano-transmitters in the solid state, then to know how to combine nano-transmitters and micro-cavities. Chapter 4, in particular, deals with semiconductor quantum boxes and their association to various types of optical micro-cavities. The chapter introduces the concepts of weak and strong coupling in micro-cavity, as well as giving reports on the applications to semiconductor lazers with a very weak threshold and to single-photon micro-nanosources required for quantum cryptography. Micro-nanostructuring of materials is also full of prospects for other active components of nonlinear optics. In fact, it is not only possible to achieve true engineering of the refraction index dispersion, but also to control the dispersion of group velocity as well as the localization of the electromagnetic field. Adapting the 2 J-M. Lourtioz, H. Benisty, V. Berger, J-M. Gérard, D. Maystre, A. Tchelnokov, Les cristaux

photoniques ou la lumière en cage, Collection Technique et Scientifique des Télécommunications, Hermès, Paris, 2003.

Preface

15

phase and group velocity of electromagnetic waves with very different frequencies in order to reinforce their interactions is an example of application in the case of second-order optical nonlinearities. Chapters 5 and 6 thus develop various aspects of nonlinear optics in micro- and nanostructured materials such as the second harmonic generation, the optical Kerr effect, the propagation of solitons or the mix of four degenerated waves. After a short theoretical introduction to nonlinear optics, the various effects are illustrated on the basis of experiments performed very recently. In Chapter 7 we openly approach the field of sub-wavelength optics with the analysis techniques of near optical field. The sub-λ nature stems not only from the distances between a point and a diffracting object, but also from fading waves whose space extension may be clearly lower than that of the light wavelength. Until recently limited to particular cases, the analysis of near fields today assumes all its interest with the development of nanotechnologies and optical micro-nanodevices. Having defined the near field concept and recalled the alternatives of microscopy in the near field, this chapter thus illustrates certain recent characterizations of semiconductor micro-components in planar integrated optics. Metallic devices involving surface or localized plasmon-polaritons are also choice objects for the studies of near fields, because these waves are not detectable in far fields. Chapter 8 is mainly dedicated to them as well as to the optical technique of microscopy by tunnel effect. The coupling between an optical wave and electric charges oscillating in a metal is a phenomenon that has been known for a long time and generally considered as a parasite, since it is dissipative over propagation lengths typically exceeding 10 microns. However, the development of micro-nanotechnologies allowed an unprecedented revival of the studies with the creation of a new set of themes known today under the name of plasmonics. The now-famous experiment of Ebbesen3 was one of the determinant elements of the renewed interest for the plasmon waves. More generally, miniaturization of metal structures appears a possible way of optical connections alongside photonic crystals. Of a smaller size than all the devices evoked previously, including quantum box nano-transmitters, nanocrystal semiconductors composed of a few hundred to a few thousand atoms belong to the category of nano-objects of great interest for small scale optics. Developed by processes different from semiconductor quantum boxes, nanocrystals can be incorporated into transparent matrices, as they can also be grafted into biological entities. Excellent candidates for the emission of “single photons”, they are also used as biological markers and present potential applications for the creation of tunable microlazers. Chapter 9 thus makes us discover the structures of the electronic levels and the optical properties of these nano-objects which, like carbon nanotubes, still remain just as attractive for the physicist. 3 J-M. Lourtioz, H. Benisty, V. Berger, J-M. Gérard, D. Maystre and A. Chelnokov (eds.), Photonic Crystals: Towards Nanoscale Photonic Devices, Springer, Berlin-Heidelberg-New York, 2005.

16

Nanophotonics

Also dealing with small-scale objects but in a very different context, the tenth and final chapter of this book completes the review of nanophotonics by addressing the interdisciplinary topic of the nanobiophotonics. The marriage of optics and biology is certainly not completely new, because while electronic microscopy offers a nanometric solution for the study of molecular cell entities, optical techniques in turn allow a slightly invasive, even non-invasive, analysis of live cells. In particular, the chapter describes the traditional fluorescence techniques for the detection of a unique molecular entity as well as more recent techniques, building on the interactions between ultra-short optical impulses and biological environments. The emergent topic of nanophotonics aims more particularly at reducing the observation volume below the limit imposed by diffraction. The chapter shows how to achieve this goal using nonlinear optical effects or nanostructured photonic devices close to the studied biological objects. The book that we have just briefly presented was written by internationally recognized specialists, each in their field. Thus, it constitutes a follow-up to the first spring school of the CNRS on nanophotonics held in Houches (France) in June 2003 and organized by the four coordinators of the book. It is, to our knowledge, one of the first times that such various and complementary aspects of nanophotonics have been gathered together. It would, undoubtedly, be useless to allot an exhaustive nature to the book, but students and scientists working in nanosciences would, however, still be able to find in it a rich source of information on the new fascinating and rapidly expanding field. Jean-Michel LOURTIOZ, Claude DELALANDE, Ariel LEVENSON, Hervé RIGNEAULT

Chapter 1

Photonic Crystals: From Microphotonics to Nanophotonics

1.1. Introduction The principal motivations for the emergence of photonic crystals can be summarized in one single word, “λ-Photonics”, which means the control of photons at the wavelength scale. The harnessing of light has always been central in the field of human endeavor: one may call to mind, for example, the destruction of the Roman fleet by the blazing mirrors of Archimedes at the siege of Syracuse in 215 BC. Generally speaking, the harnessing of light consists of structuring the space where it is meant to be confined; but there are intrinsic limitations which are related to the undulatory nature of light and which have been formulated in the famous Maxwell equations of 1873, providing a consistent picture of the experimental data then available. These limitations lie at the heart of λ-Photonics, whose definition could be the control of photons within the tiniest possible space over the longest possible time: this implies structuring space at the wavelength scale, which is the sub-micron range for the optical domain. Microphotonics or Nanophotonics? The reader will have understood that the word “λ-photonics” refers simply to Microphotonics in the optical regime. We will use the term Microphotonics throughout this chapter, given that the average size of photonic structures under Chapter written by Pierre VIKTOROVITCH.

18

Nanophotonics

consideration is micro-metric. Yet, the size resolution to be considered for the design as well as for the fabrication of efficient practical photonic devices is the nanometer. The nanometric control of the size of microphotonic structures is dictated by the specifications which are required for the resolution (better than one nanometer) of the operation wavelength; these constraints may be partly relaxed by the use of appropriate trimming procedures for the fine adjustment of the operation wavelength (control of the temperature, for example). But serious consequences may arise from insufficient control over the size of photonic structures, such as the unwanted mutual coupling of optical modes, and, as a result, the loss of control of photons within a confined space for the time required. For example, nanometric resolution of the size is essential for the production of resonant photonic structures with high quality factors, that is whose bandwidth is in the nanometer range or below (this terminology will be made familiar to the non-specialist reader in the course of this chapter). Finally, the relevance of the nanometric scale in active devices is twofold: first in connection with the quantum size active material (quantum wells and quantum boxes), and second, given the required resolution of its spatial localization within the microphotonic structure, whose design is meant to result in the ad hoc electromagnetic environment. After this parenthesis, which, beyond semantic considerations, is meant to provide to the reader an accurate definition of the “à la mode” Micro- and Nanophotonics terms, let us resume with this introductory section. A photonic crystal is a medium whose optical index shows a periodical modulation with a lattice constant on the order of the operation wavelength. The specificity of photonic crystals inside the wider family of periodic photonic structures lies in the high contrast of periodic modulation (generally more than 200%): this specific feature is central for the control of the spatio-temporal trajectory of photons at the scale of their wavelength and of their periodic oscillation duration. It will be shown in section 1.2 that there are a variety of ways of structuring space, consisting of preventing the propagation of photons along one or several directions, thus resulting in photonic “objects” with reduced “dimensionality” and with photonic properties which are strongly wavelength dependent. The new avenue opened up by photonic crystals lies in the range of degrees of freedom which they provide for the control of photon kinetics (trapping, slowing down), in terms of angular, spatial, temporal and wavelength resolution. One-dimensional photonic crystals (1DPC), which possess most of the basic physical properties of photonic crystals in general, will be discussed in section 1.3.

Photonic Crystals: From Microphotonics to Nanophotonics

19

We will embark in section 1.4 on a short and rather frustrating trip into the elusive world of three-dimensional photonic crystals (3DPC). Sections 1.5 and 1.6 will concentrate on two-dimensional photonic crystals (2DPC), which have been the subject, so far, of most new applications in terms of device demonstrations: along these lines will be presented the essential building blocks of Integrated Photonics based on 2DPC, which is presently considered as the principal domain of applications of photonic crystals. The concepts of 2.5 D Microphotonics based on 2DPC, which can be considered as a major extension of planar technology through exploitation of the third (“vertical”) dimension, will be covered in section 1.7. This chapter provides a vision complementary to that given in the book published by J. M. Lourtioz et al. (LOU 05). Concerning conceptual aspects, the present approach is more phenomenological and does not leave much room for theoretical models of photonic crystals. Particular attention is given to the changes induced by the photonic crystal on the spatio-temporal characteristics of photons immersed in the periodic medium and on similarities with phenomena observed in the case of more traditional structuring of space. As for application aspects, the present work is mainly oriented toward integrated Micro-nanophotonics: it is shown, in particular, how recent developments of 2DPC, along planar technological schemes, open the way to the production of essential building blocks for this purpose.

1.2. Reminders and prerequisites1 1.2.1. Maxwell equations The undulatory nature of light is expressed in terms of an electromagnetic field whose electrical E (r , t ) and magnetic H (r , t ) components, which depend on time t and space r coordinates, are given by Maxwell equations. The latter can be reduced to the so called master equation, as expressed below (in the case of an isotropic and non-absorbing medium): ⎞ ⎛ω ⎞ ⎛ 1 ∇ × ⎜⎜ ∇ × H (r ) ⎟⎟ = ⎜ ⎟ × H (r ) , with: r ( ) ε ⎠ ⎝c⎠ ⎝ 2

(1)

1 Section 1.2, which is a very brief and specific summary of the abundant related literature, does not include any specific references.

20

Nanophotonics

⎛ − ic ⎞ ⎟⎟∇ × H (r ) , and, E (r ) = ⎜⎜ ⎝ ωε (r ) ⎠

H (r , t ) = H (r )e jωt

(2)

E (r , t ) = E (r )e jωt

where ω is the pulsation, ε (r ) the dielectric function of the medium and c the light velocity in vacuum. This is typically an eigenvalue/eigenvector problem. 1.2.1.1. Optical modes Optical modes are the eigensolutions of Maxwell equations which correspond to a spatial distribution of the electromagnetic field which is stationary in the time scale. 1.2.1.2. Dispersion characteristics These are given by the equations which relate the pulsation (eigenvalue) of optical modes to their propagation constants (eigenvector).

1.2.2. A simple case: three-dimensional and homogeneous free space This is the simplest case, where the dielectric constant is invariant with space coordinates: the eigensolutions or eigenmodes of Maxwell equations are plane waves, with a continuous transitional symmetry. The magnetic field (as well as the electric field) can be expressed as follows: H k ( r ) = H 0 e i ( k .r ) ,

where k is the wave-vector or the propagation constant, with k =

(3) 2πn

, n is the λ optical index of the medium, and λ is the wavelength. The dispersion characteristics can be simply written as below:

ω=

c k n

(4)

c is the phase velocity of the optical mode; in the simple case considered here of n homogeneous free space, the phase velocity coincides with energy or group velocity dω . vg = dk

Photonic Crystals: From Microphotonics to Nanophotonics

21

It is shown that the density of optical modes per volume unit and per ω unit is a continuous function of ω and is written: dN ω 2 n 3 = dω π 2 c 3

(5)

1.2.3. Structuring of free space and optical mode engineering Plane waves (eigensolutions of Maxwell equations in a homogeneous medium), having a theoretically infinite spatio-temporal extension, are of no practical use from the point of view of Microphotonics, whose definition we recall: the control of photons within the tiniest possible space over the longest possible time (or, at least, over the minimum required time interval). In other words, Microphotonics is nothing but optical mode engineering, or free space carving art, in such a way that optical modes with the appropriate spatio-temporal configuration are generated. Lifetime or coherency time and quality factor of an optical mode According to the above definition of Microphotonics, it appears natural to grant the optical mode a merit factor F, which quantifies the properties of the optical mode in terms of the ratio of time τ during which it remains under control (or its lifetime from the observer/user viewpoint), over the average real space volume which it fills during its lifetime. To put it differently and more precisely, the lifetime τ is the time interval when the user may count on a coherent mode, whose phase remains deterministic, within the volume where he tries to control and confine it. The merit factor can be made T dimensionless if normalized to the ratio 3 , where T is the period of oscillation and

λ is the wavelength in vacuum. F=

Q=

τ T

×

λ3 V

λ

(6) , with:

2πτ , T

where Q is the traditional quality factor of the mode.

22

Nanophotonics

It can be shown, straightforwardly, that the merit factor F of a plane wave is nL , where n is the optical index of the medium and L is the zero. Indeed, τ = c length of the wave packet processed by the user, given that the volume V = L × S of the optical mode diverges with the section S of the plane wave. The finite lifetime of the optical mode results in a spectral widening δω =

2π

τ

.

In the simple case of a plane wave we find:

δω =

c 2π c × = δk n L n

(7)

2π expresses the “de-localization” of the mode in the wave-vector or L reciprocal space as a result of the “localization” of the plane wave along its path over a distance L. It may be noted that equation (7) is obtained from a simple differentiation of the dispersion equation (4) of the mode (plane wave here). Therefore, although time independent and formally applying in purely stationary conditions, dispersion characteristics can provide concrete information relating to the dynamics of optical modes; we will resume this discussion regularly throughout this chapter.

where δk =

Returning to the merit factor, the reader will have noticed that F is proportional to the Purcell factor, which gives the relative increase of the spontaneous recombination rate of an active medium as a result of its coupling to the optical mode, as compared to the non-structured vacuum (see Chapter 4).

1.2.4. Examples of space structuring: objects with reduced dimensionality Let us examine a few examples of space structuring, which aims in general at the production of objects with reduced dimensionality, that is where the propagation of photons is not free in all directions at any time. 1.2.4.1. Two 3D sub-spaces The simplest example of space structuring consists of dividing 3D free space into two 3D sub-spaces with different optical indices n1 and n2 , and bordered by a common infinite plane boundary (Figure 1.1).

Photonic Crystals: From Microphotonics to Nanophotonics

23

1

n1

k ⎪⎮ n2

2

Figure 1.1. Two 3D sub-spaces with different optical indices

This system can still be simply described using plane wave type optical modes, but their propagation is bound to meet with the law of Descartes at the interface, which can be simply expressed in terms of the continuity of the wave-vector component k parallel to the interface plane. Hence, if n2 ≥ n1 , plane waves propagating in medium 2 are subjected to total internal reflection at the interface for angles of incidence lying beyond a certain limit, and are evanescent in medium 1. This effect can be described in terms of the so called “light-line”, which is a straight c line whose equation can be written ω = k in ω (k ) coordinates: this light-line n1 delineates a cone within which (ω , k ) couples are allowed inside the two media, which is where refractive coupling (communication) between plane waves of the two media is allowed (see Figure 1.2). c k ) corresponds to n1 the continuum of modes which are allowed in both sub-spaces, whereas the gray zone in Figure 1.2 is restricted solely to optical modes of sub-space 2.

In other words, the inner of the cone (defined by equation ω =

1 Refractive coupling between the two sub-spaces

ω

2

c/n1

c/n2

2

k⎪⎪ Figure 1.2. Light line and light cone in ω (k ) coordinates

24

Nanophotonics

1.2.4.2. Two-dimensional isotropic propagation: planar cavity Figure 1.3 shows a schematic side view of an ideal planar cavity (thickness D) formed between two fully reflecting and loss-less metallic plane mirrors.

D

k⊥ k‫׀׀‬

Figure 1.3. Planar cavity formed between two perfect metallic plane reflectors

Propagation is allowed along the sole directions parallel to the mirrors, with the propagation vector k . Unlike in 3D free space, there is no longer a continuum of modes in ω (k ) coordinates; one observes instead a “quantification” which pπ , which are required in D order to meet the conditions for phase conservation after a “vertical” round trip of photons.

manifests itself by the discrete values of the vector k ⊥ =

The general shape of the dispersion characteristics is shown in Figure 1.4.

ω

Zero v g

cpπ/D Forbidden zone k‫׀׀‬

Figure 1.4. General shape of dispersion characteristics of an ideal planar cavity

Photonic Crystals: From Microphotonics to Nanophotonics

25

The reduction in dimensionality results in clear disturbances as compared to the biblical simplicity of the case of 3D homogeneous free space. One observes, first, a forbidden frequency zone where the density of modes is zero, below the pulsation πc ω = . In addition, curves exhibit a minimum for k = 0 ; this means that the D propagation velocity of the energy or the group velocity vg of photons tends to zero (whereas the density of modes remains finite and can be expressed as in a twodimensional homogeneous medium): the optical wave is “stationary” and oscillates “vertically”. It can be shown that the merit factor F of optical modes tends to zero, in a similar way to the case of plane waves in a homogeneous medium, except for the vicinity of k = 0 , where F tends to a finite limit which is proportional to 1

2. This fact can be interpreted as the vertical confinement of photons imposed D by mirrors resulting in a relative lateral “confinement” of optical modes, which cannot propagate any more for k = 0 .

1.2.4.3. One-dimensional propagation: photonic wire The dimensionality can be further reduced when photons are compelled to propagate along a single direction, for example inside an ideal loss-less metallic sheath (see Figure 1.5). One observes, similarly to the previous case, a forbidden frequency band below a cut off frequency where the group velocity tends to zero for k = 0 , and where it can be shown that, in addition, the density of modes diverges when the length L of the optical wire tends to infinity. Perfect metallic sheath

ω

vg = 0 and dN/dω = ∞

Forbidden zone (cut-off) k‫׀׀‬

Figure 1.5. One-dimensional propagation in a perfect metallic sheath

2 A full demonstration is not given here, due to limitations of space. It is based (as shown in the simple case of a plane wave) on the differentiation of the dispersion characteristics around the minimum. One finds, in particular, that the lateral extension of the optical mode during its lifetime τ is written S ≈ ατ , where α is the curvature (second derivative) at the extreme (minimum here) of the dispersion characteristics. Later in this chapter, we will comment extensively on the importance of the parameter α in photonic crystals.

26

Nanophotonics

The confinement strengthening of optical modes manifests itself by a finite merit factor F for finite k ( F is proportional to the inverse of the wire section S ) and by the divergence of F for k = 0 , when L tends to infinity ( F is proportional to

L ). S

1.2.4.4. Case of index guiding (two- or one-dimensionality) The guiding of photons is achieved in a more conventional way by trying to confine them inside material with an optical index higher than that of the surrounding medium: this is the principle classically applied in optical fibers. The dispersions characteristics of waveguided modes share some similarities with previous cases, but they also have notable differences. First, optical modes can be kept guided as far as they are fully prevented from communicating with the surrounding medium: their dispersion characteristics are therefore confined within the area located below the light-line (see Figure 1.6). Second, confinement is not as strong as with metallic mirrors or sheaths (an evanescent portion of the electromagnetic field is allowed to extend outside the high index material): the result of this is that the cut off phenomenon is not observed (for symmetrical guiding structures), nor does the group velocity vanish.

c/n

cc ω

No more zero vg or forbidden zone

Light-line

n k‫׀׀‬ Figure 1.6. Case of index guiding: light-line

1.2.4.5. Zero-dimensionality: optical (micro)-cavity Propagation of photons is now prevented in all directions: they are trapped in an optical cavity which can only be accessed by “resonant” modes for discrete frequencies. The spectral density of modes tends therefore to infinity at the

Photonic Crystals: From Microphotonics to Nanophotonics

27

resonance frequencies (yet, the average density of modes is finite and is not changed with respect to its free space value).

Localized modes and forbidden zones

Optical cavity

dN/dω

ω Figure 1.7. Optical microcavity: the density of modes is a series of Dirac functions at resonance frequencies

The merit factor of optical modes of an ideal cavity is naturally infinite, photons being confined in a finite volume for a theoretically infinite amount of time. In practice, the life of optical modes in the cavity is not infinite as a result of various “optical loss” processes: for example, a real metallic cavity will end up absorbing 1 photons after a finite amount of time. A spectral widening δω = of the resonance

τ

is then observed and the density of modes

1

δω

= τ remains finite. When the optical

“cage” is opened up to the external world, spectral widening extends in such a way as to overlap with the neighboring modes, and one finds again free space 3D continuum. 1.2.5. Epilogue At the present stage, the reader should be in a comfortable position to penetrate the world of photonic crystals quite easily, whose basic ingredients have already been introduced (forbidden bands, resonance, slowing down of photons). The reader may be wondering why a periodic and high index contrast structuring of space should be at all useful in the field of photon confinement. This is the essential question that we will now try to address, given that the principal ingredients of the response can be summarized in one sentence: photonic crystals provide us with new

28

Nanophotonics

degrees of freedom for the control of the kinetics (trapping, slowing down, optical losses) of photons, especially in terms of angular, spatial and spectral resolutions.

1.3. 1D photonic crystals A one-dimensional photonic crystal is a one-dimensional virtual medium, whose optical index shows periodic modulation (and it should not be viewed as a real object with reduced dimensionality in the real 3D world, as described in section 1.2.4).

1 2 ………j…………..N

r

Figure 1.8. 1D photonic crystal: the direction of propagation is normal to the layer plane

This imaginary configuration corresponds approximately to the practical situation of a periodic layer stack whose lateral dimensions are large as compared to its spatial period, given that the only relevant direction of propagation is normal to the layer plane. Dielectric layer stacks have been around for quite some time and have been widely used and developed in optics during the last few decades to control optical signals. Modeling tools are available for the design of these periodic structures, which are based on the resolution of Maxwell equations using the so called matrix transfer technique (MAC 86). It consists of the determination, step by step, of the reflected and transmitted components of the electromagnetic field at successive interfaces and within different layers. The Bragg mirror is a famous example of such a stack layer: it consists of a periodic stack of quarter-wavelength dielectric layers, with a different optical index. It is found that this structure behaves like a mirror when operating at the configuration wavelength: reflectivity increases with the number of pair layers, and for a given number of layers, with the optical index contrast between adjacent layers. The bandwidth of the reflector also increases as a function of the index contrast. For an “infinite” number of pairs, there exist spectral bands where the propagation of photons is forbidden in the periodic medium or photonic band gaps (PBGs). The concept of PBG is therefore a pretty familiar one in the world of optics and cannot be considered as a novelty introduced by the promoters of the photonic crystal concept. The novelty lies instead in the new

Photonic Crystals: From Microphotonics to Nanophotonics

29

viewpoint and the novel combinations brought about when regarding periodic structures as photonic crystals, that is by simply considering a photonic crystal in a similar way to crystalline materials: this is the so called solid state physics approach, with its wealth of generic concepts (YAB 1987; JOH 1987; JOA 1995). In the following paragraphs, we will apply this approach to the analysis of 1D photonic crystals and will see that they possess most of the basic physical properties of photonic crystals in general.

1.3.1. Bloch modes Eigenmodes of Maxwell equations in a periodic medium possess discrete periodic transitional symmetry properties. According to the Bloch theorem, these modes, also called Bloch modes, can be expressed as follows: H k ( r ) = e i ( k .r ) u k ( r ) uk (r ) = uk (r + a)

(8) with,

uk (r ) can be expanded in Fourier series: uk (r ) =

∑c

k , me

imbr

(9)

,

m

where b =

2π is the base vector of the so called reciprocal lattice. a

The essential properties of Bloch modes are summarized below: 2π ( m is an integer) are a equivalent: this is simply the mathematical expression of the diffraction process. As a result, dispersion characteristics (or photonic band structures) ω (k ) can be fully represented in the so called first Brillouin zone (according to solid state physics

– Two Bloch modes whose k vector difference is m

terminology), in the k vector range −

π a

≤k≤

π a

.

– Fourier components also express the diffraction processes induced by the photonic crystal’s periodic structure. It will be shown that they also hold the information regarding diffractive coupling properties of the relevant Bloch mode with other modes (for example, the “radiating” properties and loss characteristics, in case of coupling with the “radiation continuum”).

30

Nanophotonics

1.3.2. Dispersion characteristics of a 1D periodic medium 1.3.2.1. Genesis and description of dispersion characteristics The genesis of dispersion characteristics is illustrated in Figure 1.9 below. ω -c/n

ω c/n

-c/n k

-π/a

(a)

(b)

π/a

k

ω

ω -2 π/a

c/n

2 π/a

π/a -π/a (c) Diffraction

k

(d) π/a Coupling

k

Figure 1.9. The dispersion characteristics of optical modes in a periodic medium are widely determined by diffraction processes and optical mode coupling properties

In the case of a homogeneous medium (Figure 1.9a), dispersion characteristics, standing for propagating and counter-propagating optical waves which ignore each c other, are plain straight lines (equation ω = ± k ). The periodic structuring of the n medium results in diffraction processes, which express themselves in terms of successive discrete translations (diffraction orders: see Figure 1.9c) of dispersion 2π : the a periodicity characteristics by integer numbers (positive or negative) of a 2π of the real space has a counterpart in the reciprocal propagation vector space. a We see from Figure 1.9c that dispersion characteristics of back and forth optical c π π waves cross over at coordinates (ω , k ) = ( p ,± p ) : as a consequence, n a a propagating and counter-propagating waves may be coupled through diffraction processes. Degeneracy occurs therefore at those peculiar coordinates, since two modes which propagate in opposite directions are supposed to coexist yet have the same (ω , k ) coordinates.

Photonic Crystals: From Microphotonics to Nanophotonics

31

This coexistence will not survive beyond the time required for the coupling between modes to be completed: the degeneracy will then be raised much more strongly as the coupling is more efficient. This effect is called “anti-crossing”, resulting in the opening of a PBG (Figure 1.9d), whose width increases with the coupling rate (or the inverse of the coupling time), which itself increases with the magnitude of the periodic modulation of the optical index (periodic “corrugation”).

PBG

ω PBG

Photonic Photonic Band Gap band gap

Reflectivity, %

100 80 60 40 20

π/a

k

0 400

600

800

1,000

Wavelength, nm

Figure 1.10. Photonic band gap (PBG) and bandwidth of a Bragg reflector

cπ π ,± ) , at the first na a Brillouin zone boundaries, correspond to the case of the Bragg reflector whose For example, anti-crossings occurring at coordinates (

optical period is set at

λ

2

. The PBG width corresponds to that of the reflector

bandwidth (Figure 1.10). At the photonic band edges, dispersion characteristics exhibit extremes where the dω group velocity tends to zero. Mode coupling results there in the formation of a dk stationary optical wave, which cannot propagate. This situation is examined in more detail in Figure 1.11, where the electric field distributions of the stationary mode at the upper and lower band edges are shown.

32

Nanophotonics

Air ω2

E

2

n2

Air

PBG

n1

ω1

Dielectric

Dielectric

n1 ≥ n2

Figure 1.11. Air band and dielectric band at the photonic band edges

One may note that electromagnetic energy is concentrated principally within the low index material at the upper band edge, and vice versa at the lower band edge: this is consistent with the fact that, for a given k, photon energy decreases for increased optical index. The upper band is usually called the “air” band, whereas the lower band is called the “dielectric” band, with reference to the typical case of a photonic crystal where the low index material is air and the high index counterpart is a semiconductor dielectric. 1.3.2.2. Density of modes along the dispersion characteristics It can easily be shown that, in a homogenous and one-dimensional medium with n as optical index, the density of modes per length and pulsation unit is constant dN 2n = . In the presence of a periodic modulation of the and is expressed as dω πc optical index, this relation is simply written as follows:

π dN = 1/vg 2 dω

(10)

The density of modes tends to infinity at the edges of the PBG. This is another manifestation of the existence of stationary waves whose slowing down results in the accumulation of electromagnetic energy. The density of modes is naturally zero inside the PBG.

Photonic Crystals: From Microphotonics to Nanophotonics

33

1.3.3. Dynamics of Bloch modes

Dispersion characteristics apply in principle to periodic structures of infinite size and in stationary regime. However, the time dimension underlies the previous analysis of dispersion characteristics, especially when it comes to the coupling dynamics of optical modes and to group velocity; in addition the time dimension is inevitable in real structures of finite size. A rigorous approach would involve solving time-dependent Maxwell equations; we do not intend to analyze this aspect in detail here: intensive work in the relevant community is needed, given the ever increasing modeling requirements demanded by recent developments in the field of Microphotonics3. We will restrict ourselves to a brief discussion of optical mode dynamics, based on simple analytical relations. 1.3.3.1. Coupled mode theory The coupled mode theory was originally proposed by Kogelnik and Shank in 1972 for the analysis of distributed feedback lazers (see also TAM 1988): this analytical theory is well suited to the modeling of microphotonic structures whose operation is essentially based on coupling phenomena between optical modes. Let us briefly recall that it is usually associated with a matrix formalism, which allows for the cascading of elementary building blocks in order to assemble more complex systems. The basic ingredients of the theory are summarized below. Coupling between two modes is described by a coupling constant κ (cm −1 ) , which depends on the magnitude of the periodic structuring of the optical index and on the overlap integral to the electromagnetic field distributions of the two modes. Lc = 1 is the

κ

coupling length: in other words, enough “time” is given for coupling to occur, n , where n is the provided that the medium size exceeds Lc . Coupling time τ c = cκ “average” optical index of the medium. The PBG opened up by coupling between two modes (see section 1.3.2) is written: PBG ≈

4π

(11)

τc

3 Note the availability of simulation tools, which make use in particular of the FDTD (Finite Difference Time Domain) method, which allows for the 3D simulation of finite size structures in the spatial, spectral and temporal domains. There is still a long way to go, however, before efficient design tools for microphotonic devices are available.

34

Nanophotonics

It can also been seen that, at the photonic band edges where the group velocity is null, the curvature α (or second derivative of the dispersion characteristics) is proportional to τ c . 1.3.3.2. Lifetime of a Bloch mode The concept of lifetime τ of a Bloch mode in a 1D structure is meaningless unless its size L is limited (in the absence of any other loss mechanism). This aspect has already been discussed in section 1.2.3: we have shown that the limited 2π lifetime of the mode results in a spectral widening δω = ; we have also shown

τ

that it is possible to relate δω (by differentiating the dispersion characteristics) to 2π the “de-localization” of the mode in the reciprocal space δk = , which results L from its “localization” in real space (the mode is allowed to extend over L). The differentiation of the dispersion characteristics around an operation point can be written:

δω = v g δk +

α 2

(δk ) 2 +

β 6

(δk ) 3 + ...

(12)

It can be expressed in terms of the lifetime of the mode within the structure of limited size as below: 1

τ

= vg

1 α 2π 2 β 2π 3 ( ) + ( ) + ... + L 4π L 12π L

We find, as expected, τ =

(13)

L , in a homogeneous medium, where the group vg

velocity v g is independent of the wavelength (and coincides with the phase velocity). In a structured medium and, in particular, around an extreme of the dispersion characteristics where the group velocity of the mode, so called slow Bloch mode, vanishes, it is found (limiting the series to the second order expansion):

τ=

L2

(14)

πα

The lifetime of the slow Bloch mode increases as L2 and is proportional to the inverse of the curvature around the extreme. Note that the smaller the curvature, or the stronger the coupling between optical modes giving rise to the extreme in the dispersion characteristics, the longer the lifetime of the resulting slow Bloch mode:

Photonic Crystals: From Microphotonics to Nanophotonics

35

the curvature α at the extreme is the relevant parameter for the description of the slowing down of photons within the periodic structure. 1.3.3.3. Merit factor of a Bloch mode The general definition of the merit factor of an optical mode (see section 1.2.3, equation (6)) results in the following relations: F=

c = ng , vg

(15)

in linear regime, where n g is called the group optical index, corresponding to the group velocity vg : F=

Lc

πα

,

(16)

at an extreme of the dispersion characteristics where the curvature is α . One therefore finds that the merit factor is independent of the length of the structure in linear regime, whereas it increases linearly with the length at the extreme, steeper as the curvature at the extreme is smaller. This manifestation of the lateral “confinement” of the mode (although de-localized) should be familiar to the reader: it was discussed in section 1.2.4, concerning the behavior of optical modes in a planar cavity around the extremes of the dispersion characteristics (for k = 0 : see footnote 2). Confinement is now achieved owing to the sole presence of periodic structuring, resulting in the existence of slow Bloch modes, enabling the build-up of electromagnetic energy in a confined space, over a long period of time: it manifests itself by a resonance in the spectral domain, arising from the presence of (slow Bloch) modes which are intrinsically de-localized, although efficiently confined in practice. 1.3.4. The distinctive features of photonic crystals

At the present stage and from the analysis in the previous section, although restricted to 1DPC, it is possible to derive the principal characteristics that make up the distinctive features of photonic crystals, in general. Let us remind ourselves that photonic crystals are strongly corrugated periodic structures (large magnitude of the periodic modulation of the optical index). This results in a strong diffractive coupling rate between optical modes and by significant

36

Nanophotonics

disturbances of the dispersion characteristics as compared to a homogeneous medium. These disturbances manifest themselves by the presence of: – large photonic band gaps (PBG); – flat photonic band edge extremes (where the group velocity vanishes) with low 1 . curvature (second derivative) α ≈ PBG These are the basic ingredients which make photonic crystals the most appropriate candidates for the production of a wide variety of compact photonic structures. “De-localized” slow Bloch modes, discussed in the previous section, which lend themselves to the production of compact resonant structures, are one example of that matter. Another example was provided by the compact and highly reflective Bragg mirror, formed with a small number of quarter-wavelength high index contrast pairs. We now come to the use of 1DPC, operating in the field of PBGs, for the production of compact resonant structures based on “localized” optical modes.

1.3.5. Localized defect in a photonic band gap or optical microcavity

Ideal 1D periodic structures have been considered so far. If the “crystalline” periodicity is broken locally, this results in the formation of a “localized” defect which manifests itself as a “localized” resonant optical mode within the PBG, in quite the same way as a crystalline defect introduces localized defect states in the band gap of crystalline semiconductor material. According to optics terminology, the localized defect is called an optical microcavity where the corresponding localized optical modes are confined. A well-known example is the Bragg mirror, where the optical thickness of one of its pairs is changed with respect to the quarterwavelength configuration, as illustrated in Figure 1.12. Cavity modes are determined by the conditions for resonance, which can be expressed as: (

2πn1

λ

) × 2 D + Φ1 + Φ 2 = p 2π ,

(17)

where Φ1,2 are the phases of mirror reflectivity: at resonance, photons do not experience any phase difference (modulo 2π ) after a round trip in the cavity. For Bragg mirrors whose configuration wavelength corresponds to the operation wavelength (around the center of the PBG), relation (17) is simply written as:

Photonic Crystals: From Microphotonics to Nanophotonics

D= p

λ

37

(18)

2n1

D

n2 e2 e1

Φ1

n1

Φ2

Figure 1.12. Optical microcavity formed between two Bragg reflectors

1.3.5.1. Donor and acceptor levels

n1 ≥ n2 Air

n1 Donor Acceptor

n2

Dielectric

Figure 1.13. Donor or acceptor type localized state Figure 1.13. Donor or acceptor type localized state

A defect can be created, for example, by increasing the high index portion of a quarter-wavelength pair: this results in a shift in air band levels towards the PBG

38

Nanophotonics

and the introduction of so called donor type localized levels in the latter. When the low index portion of the pair is widened, the localized levels are acceptor-like, and originate from the dielectric band. 1.3.5.2. Properties of cavity modes in a 1DPC A defect in a 1DPC is an object with 0 dimensionality in 1D space. Properties of localized optical modes in an 0D object have already been discussed in section 1.2.4. The spectral density of modes tends to infinity at resonance wavelengths; this is also true for the merit factor F, in the absence of any optical loss processes, which would limit the optical mode lifetime τ (which is also infinite in the absence of losses). This is not true in practice: for example, for a cavity formed between two Bragg reflectors with finite thickness, optical losses arise from the escape of photons across the mirrors whose reflectivity R is lower than 1. The lifetime of optical cavity modes and their merit factor can then be expressed as:

τ≅

1 nD , × 1− R c

F≅

n , 1− R

(19)

where n is the optical index of the cavity material. The density of modes is no more infinite at the resonance wavelength and the spectral response is widened as shown in Figure 1.14. n D R

dN/dω ≈τ δω ≈

cπ nD

1

τ

= (1 − R )

R

cπ nD

ω

Figure 1.14. Optical mode density in a cavity formed between two reflectors

Photonic Crystals: From Microphotonics to Nanophotonics

39

nD (the inverse of the free spectral range cπ between two successive cavity modes) and increases with cavity size (it is constant n and equals , when expressed per length unit): the number of cavity modes cπ increases therefore with size, in a given spectral range.

The average density of modes is

For R = 0 , that is with no reflectors, optical mode density at resonance coincides with the average density of modes (which itself coincides with that of a homogeneous medium), and spectral widening corresponds precisely to the free spectral range between two cavity modes: we are back to the state continuum in 1D homogeneous free space. 1.3.5.3. Fabry-Pérot type optical filter An optical cavity formed between two Bragg reflectors with finite thickness behaves like a wavelength selective filter. An incident plane wave is essentially reflected except for the resonant wavelength of the cavity, where it may couple with the cavity modes and be, at least in part, transmitted across the structure. If the latter is symmetrical (identical reflectors), the transmission can reach 100% at resonance wavelengths. The selectivity of the filter (spectral width of the transmission spectrum) is equal to the spectral widening δω , which is related to the finite lifetime of the cavity mode (see equation (19) and expression of δω , Figure 1.14). This type of device is usually called a Fabry-Pérot cavity filter. Filter selectivity is therefore controled directly by the reflectivity of the Bragg mirrors (in the absence of any other source of optical losses). Wavelength tuning of the filter can be achieved simply by changing the optical thickness of the cavity (that is its physical thickness and/or its optical index). Use of photonic crystal-type Bragg reflectors, formed with high index contrast pairs, enables the production of strongly resonant, yet extremely compact, structures: a limited number of quarter-wavelength pairs is required to produce high reflectivity Bragg mirrors (SPI 98). Filter # L03B08n°2 PO

Transm ission (dB)

0

0V

-5

15 V

-10

24 V 30 V

-15

32 V

-20

33 V

-25 -30 -35 -40 1375

1385

1395

1405

1415

1425

1435

1445

1455

W avelength (nm )

Figure 1.15. Micrograph and spectral response of a tunable Fabry-Pérot filter formed with high index contrast air-semiconductor pairs (Collaboration LEOM-ECL-CNRS/ATMEL)

40

Nanophotonics

Figure 1.15 shows an example of a Fabry-Pérot filter, formed with high index contrast air/semiconductor membrane pairs. This filter can be tuned by changing the thickness of the air cavity layer via electrostatic actuation. As to the selectivity of the filter, Bragg mirrors with only two quarter-wavelength pairs are sufficient to achieve a spectral bandwidth of around one nanometer.

1.3.6. 1D photonic crystal in a dielectric waveguide and waveguided Bloch modes

A 1D photonic crystal does not actually exist, in the same way that a 1D space does not exist. A closer representation of the real world would be to imagine, in 3D real space, a 1D structuring, where one would consider the sole plane waves having a propagation vector along the direction normal to the iso-optical index planes. This implies an infinite structuring depth, which is not realistic either. An additional step towards reality consists of considering a 1D structuring of a 2D object in real 3D space. The planar dielectric waveguide, where photons are so called “index guided”, already presented in section 1.2.4, is a well-known example of a 2D object. If we now admit, as a final step in non-reality, that the lateral size of the 1D structuring is infinite and that the only considered propagation is parallel to the periodic index gradient, it is possible to represent this situation with a 1D propagation in a 2D world. The dispersion characteristics ω (k ) of optical modes in a planar waveguide free of structuring were presented in Figure 1.6. In the presence of a 1D periodic structuring, dispersion characteristics are deeply modified, as a result of a variety of coupling processes which affect the propagation of optical modes in the dielectric waveguide. The corresponding eigen waveguided modes are also called Bloch modes, and their symmetry properties along the direction of propagation are similar to those presented in the case of an ideal 1DPC (see section 1.3.1). 1.3.6.1. Various diffractive coupling processes between optical modes We observe, first, diffractive coupling between propagating and counterpropagating waves, which may now communicate as described in section 1.3.2 in the case of an ideal 1DPC, and the resulting effects on dispersion characteristics (photonic band gap and band edge extrema, etc.). A second essential consequence of diffractive processes lies in the new channels opened up to waveguided modes for communication with radiated modes in 2D free space: this may occur as soon as the discrete translation of the dispersion characteristics induced by diffraction can shift them, at least partially, above the light-line. This can be represented by a simple geometrical operation consisting of the successive folding of dispersion characteristics around the vertical axis of equation k = ±

π

a gathering them within the first Brillouin zone (see Figure 1.16).

and, consequently,

Photonic Crystals: From Microphotonics to Nanophotonics

c

c

41

c/n

ω

a π/a

k‫׀׀‬

Figure 1.16. 1D photonic crystal in a dielectric waveguide: coupling processes between propagating and counter-propagating waves, and between waveguided and radiated modes

In these conditions, loss-less and pure waveguided modes correspond only to the portions of the dispersion characteristics lying within the triangular area (whose three sides are drawn in bold in Figure 1.16). For the other portions located above the light-line, the waveguided modes are “lossy”, which means that their lifetime in the waveguided state is finite: they will end up lost in the radiation continuum of the free 2D space. Note that the room available for (ω , k ) couples corresponding to pure waveguided modes is much larger as the effective index of the guide is higher, with respect to that of free space. Finally, there exists a further diffractive coupling process which allows for communication between different order waveguided modes corresponding to different dispersion characteristics. These waveguided modes would otherwise ignore each other in a non-structured waveguide, since they represent eigen orthogonal solutions of Maxwell equations (applied to the uniform non-periodic waveguide). Two conditions should be met in order for the coupling to occur. First, the diffraction condition must be satisfied, that is to say that, following the translation of the relevant dispersion characteristics, there exists a crossing point, which determines the couple (ω , k ) where the coupling can be achieved (Figure 1.17). Second, symmetry conditions have to be met, in such a way that the overlap integral to the electromagnetic distributions of the two modes, weighted by the spatial distribution of the periodic index, is not zero. For example, if the optical index vertical distribution is even, then the sole modes with identical (even or odd) vertical symmetry will be able to couple each other (even modes in the example given in Figure 1.17). These mode coupling processes result in the opening up of new PBGs with new photonic band edge extrema, which do not necessarily coincide with the Brillouin zone boundaries, that is in the existence of slow Bloch modes at

42

Nanophotonics

extrema of the dispersion characteristics which do not necessarily coincide with high symmetry points ( k ≠ ± p

π

a

).

Light-line

ω

LL

en 3 - ev Mode d - od e2 d o M even e 1d o M

ω Radiation modes

k k

π/ a

2π/ a

Figure 1.17. Coupling between waveguided modes of different orders (here with the same even symmetry: the only crossing point where coupling is possible is shown in the circle)

1.3.6.2. Determination of the dispersion characteristics of waveguided Bloch modes The reader will have concluded that the dispersion characteristics or band structure of a 1DPC formed in a planar waveguide are relatively complex, given that, in addition, one obtains different characteristics for the two TE (in plane electric field and transverse magnetic field) and TM (vice versa) polarizations. Different methods of band structure calculation are available. A simple approach, called the effective index method, provides satisfactory results: it consists of the use of an effective index for each mode (evaluated in the non-structured waveguide) and in the resolution of Maxwell equations in the periodically structured space, with the appropriate dimension (1D, for example, in the present case). One may also consider a more rigorous method, the so called plane-wave method with “supercell”, which is used frequently (VAS 02): this consists of the resolution of Maxwell equations in a space (here 2D), periodically paved with elementary cells of rectangular symmetry, which include the waveguide and a portion of the surrounding space: the period along the direction of propagation coincides with that of the PC structuring, whereas it is large enough in the normal direction to consider that the electromagnetic field is negligible at the boundary of the supercell. This method is well suited to describing waveguided modes below the light-line.

Photonic Crystals: From Microphotonics to Nanophotonics

43

1.3.6.3. Lifetime and merit factor of waveguided Bloch modes: radiation optical losses The dynamics of guided Bloch modes below the light-line are very similar to those described in section 1.3.3 for the simple case of an ideal 1DPC: they can also be described using the coupled mode theory, and using dispersion characteristics. One may, in the same way, differentiate the dispersion characteristics in order to determine the mode lifetime and their merit factor as well, in a structure with a limited size L, thus ending up with relations (13) – (16): we remember, therefore, that their lifetime (their “time of flight”) within the structure is an increasing function of L (increases like L in linear regions of the dispersion characteristics, like L2 at extrema with finite curvature). As to Bloch modes which may couple to the radiated continuum, that is to say whose dispersion characteristics can be “folded” above the light-line by diffraction processes, their lifetime will be necessarily limited to a maximum value, which is their coupling time constant τ c with radiated modes. The lifetime of Bloch modes “above the light-line” can then be written: 1

τ where

1

τg

=

1

τg

+

1

τc

,

(20)

is given by the general relation (13), which expresses the lifetime in

absence of coupling with the radiated continuum. We emphasize here the radiation optical losses that may significantly affect PC-based devices for integrated optics, which are meant to operate in the waveguided regime. Another way of formalizing these effects is to define the minimum distance Lc which can be explored by photons before being lost as a result of radiation losses. One find simply Lc = v gτ c in linear regions of the dispersion characteristics and Lc = πατ c at extrema, in the present case of a 1DPC formed in a planar waveguide (see equation (13)). It must be pointed out that around an extremum where the Bloch modes are slowed down, this distance can be strongly reduced in a PC, where the curvature α at an extremum can be made very weak, as discussed in section 1.3.4. This remark is of major importance, if one considers the reverse situation where one couples radiated modes to waveguided modes, that is to say when one injects photons in a waveguide from free space via diffractive coupling: it is clear that, with a PC, this function can be achieved in a very compact way, unlike with traditional diffraction gratings; we will return to this point in more detail later in the chapter (section 1.7.1).

44

Nanophotonics

1.3.6.4. Localized defect or optical microcavity As in the case of an ideal 1DPC, the introduction of a defect in a 1DPC formed in a planar dielectric waveguide can result in an optical microcavity: it consists here of an object with 0 dimensionality, formed in a 1D structure (waveguided 1DPC), considered itself in a 2D space. A schematic view of the microcavity is shown in Figure 1.18, together with the corresponding dispersion characteristics: the latter are simply reduced to a horizontal straight line, which corresponds to the resonance frequency of the cavity. This results from the fact that the defect, being localized in real space, is de-localized in the k vector reciprocal space and that all k components are available for the cavity modes: a localized defect can therefore potentially couple with any other mode and, in particular, with the modes of the radiation continuum, using the k components corresponding to points of dispersion characteristics located above the light-line (Figure 1.18).

Light line Light-line

a

ω

D

k‫׀׀‬ Figure 1.18. Dispersion characteristics of a localized defect or of an optical cavity: there always exist components of the k vector located above the light-line

The coupling rate between the defect and another mode depends naturally on the distribution function of the k Fourier components; this is true for coupling with radiated modes, which is determined by the proportion of the Fourier components located above the light-line and defining the so called radiation diagram of the defect. It is possible, in principle, to design a defect in such a way as to control the

Photonic Crystals: From Microphotonics to Nanophotonics

radiation diagram4 in order, for example, to minimize the distribution of k

45

vectors

above the light-line and, therefore, to reduce optical losses in free space. Coupling of the cavity modes with the radiation continuum tends to reduce their lifetime, which results in the spectral widening of their resonance. Let us return to the simple case of the Fabry-Pérot cavity formed between two Bragg reflectors of limited size (which is equivalent to creating a defect in a 1DPC with limited extension) – but this time in the waveguided configuration (Figure 1.19). The 1 of photons lifetime τ of cavity modes is not only limited by the escape rate

τR across the Bragg reflectors (derivable from relation (19)), but also by the coupling rate

1

τc

with the radiated modes; it is therefore given by the relation: 1

τ

= (1 − R )

(21)

1 c + nD τ c

n D R

R

a

D

Figure 1.19. Microcavity or defect in a 1DPC, formed in a dielectric waveguide

The structure described schematically in Figure 1.19 may also be used as an optical filter in waveguided configuration. The art of the designer lies in his ability to propose configurations where optical losses are kept to a minimum, resulting in strongly resonant microcavities, useful for the production of very selective filters, as well as for the control of spontaneous emission in an active medium. We will return to this point in the section devoted to 2DPC (section 1.6.2).

4 Remember that the distribution of wave-vectors and the spatial distribution of the electromagnetic field relate to each other through a Fourier transformation.

46

Nanophotonics

1.3.7. Epilogue

We have reached the end of our discussion of the basics of one-dimensional photonic crystals. The principal concepts which hold for photonic crystals in general have been presented. The extra ingredient brought about by photonic crystals with a dimension larger than one lies essentially in the extra control of light which they may provide in terms of angular resolution. The illustration given in Figure 1.20 provides a classic example of the limitations of 1D photonic crystals with respect to angular resolution: a Bragg mirror does not usually have the capacity to fully reflect the light in a given wavelength spectral range for any angle of incidence. The rest of this chapter is devoted to photonic crystals with dimensions larger than one, principally 2DPC (sections 1.5, 1.6 and 1.7), after a very brief trip into the world of 3DPC (section 1.4). R

λ

Figure 1.20. A limitation of Bragg mirrors: their transfer function depends on the angle of incidence, which usually prevents the formation of a PBG for all propagation directions

1.4. 3D photonic crystals 1.4.1. From dream …

R

λ

Figure 1.21. A 3D photonic crystal may behave as an omni-directional perfect reflector in a finite spectral bandwidth. As shown in the figure, there exists a range of wavelengths (crosshatched area) where reflectivity is uniform, for several directions of propagation

Photonic Crystals: From Microphotonics to Nanophotonics

47

A 3D photonic crystal’s optical index shows high contrast periodic modulation along the three dimensions of space. The main consequence lies in the new opportunity given to the eigenstates of three dimensional space to experience strong mutual coupling via diffraction processes, whereas the plane waves, eigenmodes of the homogeneous space with continuous transitional symmetry, ignore each other. Here we recall all the concepts and specifications of 1DPC analyzed in detail in section 1.3, but this time along the three dimensions of real space. As a celebrated example (this is the usual way in which 3DPC is introduced, though it is very restrictive), a 3D PC may show a full PBG for all directions of space, which means that it behaves as a perfect reflector for all angles of incidence, as illustrated in Figure 1.21. The concept of 3DPC was introduced in 1987 by E. Yablonovitch (YAB 87) and demonstrated experimentally for the first time in the field of microwaves in a photonic crystal called “Yablonovite” (YAB 91). The initial motivation was to gain full control of the spontaneous emission of an active emitting material: for example it can be fully inhibited if the active material is inserted in a photonic crystal showing a regime of PBG in a spectral range larger than the emission spectrum of the material in homogeneous vacuum; the active material may also be inserted in an optical cavity resulting from the formation of a defect in the 3DPC (under the same principle as in the case of a 1DPC), and be left the limited choice of a few well controled (numerical and spectral characteristics) cavity modes to take care of radiated recombination processes. In the field of Micro-nanophotonics, the quest for the “grail”, motivated by new exciting developments in the fields of Quantum Electrodynamics and of Quantum Information (very low threshold lazer micro-sources, strong coupling: see Chapter 4), consists of inserting “a quantum-box” type nano-emitter in a “photonic box” or microcavity, which would accept only one optical mode in the emission spectrum of the emitter, with a perfect spectral overlap. 3DPC are potentially the best candidates for those purposes. Other brilliant applications can be contemplated, in principle, such as 3D loss-less perfect guiding of photons, by opening up a corridor across a forbidden zone, etc. 1.4.2. … to reality

The fabrication technology of 3DPC in the optical domain is extremely complex. The only examples of mass production of such structures can be found in nature, owing to the miracles (or to the 500 millions years of research) of natural morphogenesis. Butterfly wings provide a brilliant example of nature’s

48

Nanophotonics

achievements in this respect: their seductive colors are produced by the interaction between ambient light and their natural micro-structuring (Figure 1.22). A great deal of technological research has been devoted to the production of artificial 3DPC. Figure 1.23 shows a few examples of recent achievements (woodpile structure, inverse opal, 3D hole matrix similar to the famous “Yablonovite” etc. – see for example (CUI 99; LOU 05) and references therein).

SEM view

Figure 1.22. An example of a 3D photonic structure: butterfly wings

Lin et al.

Cuisin et al.

Campbell et al.

Figure 1.23. Examples of artificial 3DPC

Photonic Crystals: From Microphotonics to Nanophotonics

49

However, 3DPC should be considered as laboratory objects; their transformation into high performance photonic devices is still some way ahead. As a consequence, most R&D efforts have concentrated on 2DPC, which is far more promising in terms of practical applications, at least in the short/medium range.

1.5. 2D photonic crystals: the basics

A two-dimensional photonic crystal is a virtual two-dimensional medium whose optical index is periodically modulated: such a medium should not be confused with an object with a dimensionality reduced to 2, in a real world of dimension 3, as described in section 1.2.4. This imaginary situation could be illustrated approximately, for example, by a dielectric material dug with a periodic lattice containing deep and parallel cylindrical holes (the depth should be much larger than the lattice constant5), the wave propagation being considered solely along directions normal to the holes. A cross-section of such a periodic lattice, with discrete triangular transitional symmetry, is shown in Figure 1.24.

80 PC rows : L~42µm

y

80 PC rows : L~36µm

a1 a a3 2 x

Figure 1.24. A cross-section of a 2DPC with triangular symmetry

5 This configuration has been realized experimentally in the so called macro-porous silicon approach (GRU 96; ROW 99).

50

Nanophotonics

For the rest of this chapter we will use the triangular lattice case, which is widely used (though not exclusively so) in practice as an illustrative example. Two of the three vectors a j ( j = 1,2,3) in Figure 1.24 allow for a complete description of symmetry properties of the 2DPC in real space, and in particular of the permittivity of the periodic medium:

ε (r + a j ) = ε (r )

(22)

1.5.1. Conceptual tools: Bloch modes, direct and reciprocal lattices, dispersion curves and surfaces

In the following paragraphs we will use the conceptual tools we used for the analysis of 1DPC, now applied to 2DPC. 1.5.1.1. Bloch modes Eigenmodes or solutions of Maxwell equations in a two dimensional periodic medium are also Bloch modes, which possess the same translational discrete periodic symmetry properties. Bloch theorem applies in the same way and the electromagnetic field is written:

H k ( r ) = e i ( k .r ) u k ( r ) uk (r ) = uk (r + a j )

(23) with,

uk (r ) can therefore be written according to a Fourier expansion: uk (r ) =

∑c

k , m, j e

imb j r

(24)

m, j

where b j is a base vector of the reciprocal lattice, described in detail in the following paragraphs, with b j = base vector of the direct lattice).

2π 2 2π × ≈ × 1.15 ( a is the modulus of the a a 3

Photonic Crystals: From Microphotonics to Nanophotonics

51

1.5.1.2. Direct and reciprocal lattices Figure 1.25 shows again the triangular periodic lattice in real space and its reciprocal representation in k vector space. ΓΜ and ΓΚ stand for the two principal so called high symmetry directions of the lattice.

∑

Any vector of the reciprocal space can be projected on the base vectors and m j b j , where m j is an integer. expressed as mj

We recall briefly the essential properties of Bloch modes, already discussed for 1DPC, but extended this time to the case of a 2DPC: – Two Bloch modes whose k vectors differ by

∑m b

j j

are equivalent: this is

mj

the mathematical translation of two dimensional diffraction processes. – Fourier components (equation (24)) include the physical reality of diffraction processes induced by the periodic structure of the photonic crystal. They also contain information about the possibility of diffractive coupling with other modes as well as about the radiating characteristics of the mode, in case of coupling with the “radiated continuum” (which we will encounter again in a real 2DPC, formed in a planar dielectric waveguide, in section 1.5.2). ky

y

a1 a2 a3

K x

ΓK

ΓM

M Γ

b

kx

ΓK, ΓM: high symmetry direction

Figure 1.25. Direct and reciprocal triangular lattices. The two so called ΓΜ and ΓΚ high symmetry directions of the crystal are shown. The first Brillouin zone is included in the hexagon drawn in the reciprocal lattice. b is the module of the base vector of the reciprocal lattice

52

Nanophotonics

1.5.1.3. Dispersion curves and surfaces It is appropriate here to speak in terms of dispersion surfaces ω (k ) = ω (k x , k y ) , real space being two-dimensional. In a non-structured homogeneous space, the dispersion surface is simply a conical surface (Figure 1.26), whose equation can be written:

ω = c k x2 + k y2 ,

(25)

where c is the light velocity in the homogeneous medium. Diffraction processes induced by high index contrast 2D periodic structuring affect the dispersion surfaces significantly, in the same way as in the case of 1DPC: one also observes the formation of photonic band gaps, whose related band edges coincide with low curvature extrema of surface dispersion characteristics. These characteristics are also the result of diffractive coupling processes between optical modes, induced by periodic structuring. Dispersion surfaces (or photonic band structures) are periodic and can be fully described (folded) within the so called first Brillouin zone according to Solid State Physics terminology, corresponding to the dark gray hexagonal surface of Figure 1.25.

ω ky kx Figure 1.26. Conical surface dispersion of a two-dimensional homogenous medium

There exist various simplified representations of dispersion surfaces, whose full 3D visualization does not allow for simple extraction of essential characteristics. One may plot, for example, iso-frequency curves (which are plain circles with

ω

c radius in a homogeneous medium), which are the cross-section of the dispersion surface with planes of a given ω coordinate. Iso-frequency curves are no longer isotropic, as in the homogeneous space, and are subject to symmetry rules dictated by the periodic lattice, in particular along the high symmetry directions. The group velocity is now written: (26) vg = ∇ kω ,

Photonic Crystals: From Microphotonics to Nanophotonics

53

and its direction, which coincides with the direction of energy propagation, is ⎛ ω ⎞ normal to the iso-frequency curves. Thus, in general, phase velocity vϕ = ⎜ 2 ⎟k ⎜k ⎟ ⎠ ⎝ and group velocity do not travel in the same direction. We will comment below on the consequences, in terms of potential applications, which may result from these non-conventional properties, especially around the extrema of the dispersion characteristics. Another widely used representation scheme (Figure 1.27) consists of plotting the dispersion characteristics along a given propagation direction of Bloch modes, obtained from the cross-section of dispersion surfaces with the plane defined by the frequency axis and the direction of propagation. This scheme is usually applied, in a concentrated yet partial manner, by restricting the propagation directions to the sole high symmetry directions ΓΜ and ΓΚ (JOA 95)6; for other directions (between Μ and K), the diagram includes only the points corresponding to the band edges or, in other words, corresponding to the Brillouin zone boundary (where optical waves propagating in opposite directions can interfere via diffractive coupling). Along high symmetry directions band edges show “full” extrema. This representation scheme is illustrated in Figure 1.27, where a full photonic band gap can be observed (for the TE polarization, gray area), at the first Brillouin zone boundary. K

M ΓΜ =

ω

BI

b 3

ΓΚ =

b 2

Γ y

a1 a a3 2

k

x ΓM

ΓK

Figure 1.27. Partial representation of the dispersion characteristics of a 2DPC. The modules of the K vectors (⏐ΓM⏐ and ⏐ΓK⏐) are given at the first Brillouin zone boundary for the high symmetry directions (b is the module of the base vector of the reciprocal space) 6 Note that this restriction may turn to an insufficient representation, especially in case of complex basic elementary structures.

54

Nanophotonics

The photonic band gap increases with the coupling rate between optical modes induced by the periodic structuring and, therefore, with the magnitude of its modulation, as already discussed in the case of 1DPC (section 1.3.2). In the case of a periodic array of holes in a dielectric matrix, the coupling rate increases with the “air” filling factor f (to the extent that the holes are fully defined and do not join); this is also true for the PBG as illustrated in Figure 1.28, where the band edge curves are shown as a function of f (the PBG area is white). Note the use of normalized coordinates a , where a is the lattice parameter of the triangular lattice, for the λ

a/λ

frequency, or energy, axis (JOA 95).

f (air %) Figure 1.28. Photonic band gap as a function of the hole filling factor (from Le Vassor D’Yerville, GES, Montpellier)

1.5.2. 2D photonic crystal in a planar dielectric waveguide

Let us recall that an ideal 2DPC (two-dimensional object in a two-dimensional world) formally has no real existence and that a representation closer to reality would be to imagine, in a real 3D space, a 2D structuring of the latter, where the only directions of propagation to be considered would be along the index “gradient vector”. This implies an “infinite” shape ratio of the structuring (holes), which is not realistic either. A last step towards reality consists of considering a 2D structuring of a 2D object in 3D real space (Figure 1.29): the planar dielectric waveguide where photons are “index guided”, that is to say vertically confined by the vertical profile of the optical index, already discussed in section 1.2.4 and exploited with 1DPC (section 1.3.6), is the well known example of a 2D object, which we will again examine below.

Photonic Crystals: From Microphotonics to Nanophotonics

55

Figure 1.29. Schematic representation of a 2DPC formed in a planar dielectric waveguide

All the concepts presented in the case of 1DPC formed in a planar dielectric waveguide essentially apply to 2DPC. One finds again (and again) diffractive coupling processes and their great impact on dispersion characteristics. Figure 1.30 shows an example of dispersion characteristics (the coordinates are normalized) of a triangular 2DPC formed in dielectric membrane similar to that shown in Figure 1.29. The light-line discussed in detail in the case of 1DPC is now a light surface or a light cone which meets the equation ω = c k x2 + k y2 , where c is the light velocity in the surrounding homogeneous medium. In the partial representation of surface dispersions given in Figure 1.30, the light cone is also partially represented by the line which separates the radiated continuum area (gray) from the area which corresponds to loss-less Bloch modes (white). The dispersion characteristics of lossy Bloch modes (which may couple with radiated modes) are not plotted.

Gap from 0.325 to 0.428

Figure 1.30. Dispersion characteristics of triangular 2DPC formed in a planar dielectric waveguide. The curves corresponding to waveguided Bloch modes below the light cone are solely represented (the area corresponding to the radiated continuum is shown in gray)

Again we encounter the concepts of lifetime and merit factor of Bloch modes and the impact of radiation losses on the latter, already discussed in section 1.3.67. 7 The concept of lateral extension of the Bloch mode is now translated in terms of the surface of the Bloch mode. For example, if one considers an operation point at an extremum of the

56

Nanophotonics

In summary, all the distinctive features and virtues of photonic crystals, already discussed in the simpler context of 1DPC and allowing for a fine engineering of optical modes, that is for their slowing down, their trapping and their spectral selection, using compact structures, may apply to 2DPC. The extra “dimension” brought by 2DPC is, however, essential: its importance lies in the considerable increase in degrees of freedom offered in terms of spatial and angular resolutions. For example, the shape of defects which can be introduced in the 2D lattice can be changed in an “infinite” number of ways: we will return to this particular aspect in the sections devoted to the building blocks of integrated photonics. Angular resolution is also a characteristic which becomes a virtue in 2DPC, opening the way to a wide range of applications, in connection for example with their non-conventional properties (dispersion, refraction), which manifest themselves at extremes of the dispersion characteristics (see for example (GRA 00)). This latter aspect is illustrated below, after a brief description of the so called super-prism effect. 1.5.2.1. An example of the potential of 2DPC in terms of angular resolution: the super-prism effect

θ1 n2

Normal refraction n 1/2 independent of ω

n1

k i1= k i2

ki

θ2

n 1cos( θ1) = n 2cos( θ 2 ) Super-refraction

kt

ki ≅ ΓΚ 2 Gap from 0.325 to 0.428

ki

Figure 1.31. Normal refraction and super-refraction

dispersion characteristics, where the curvature is α and assumed isotropic, it can be shown easily that the maximum surface attainable by the corresponding “slow” Bloch mode is 1 is the radiation loss ratio. written S BM ≈ ατ c , where

τc

Photonic Crystals: From Microphotonics to Nanophotonics

57

“Normal” refraction, described in section 1.2.4, manifests itself by a deviation in the direction of propagation of a plane wave, when crossing the interface plane between two media with different optical indices n1, 2 : this deviation is dictated by the law of Descartes, which simply expresses the conservation of the component of the propagation vector parallel to the interface plane. As a result, the angle of incidence θ1 of the wave originating from medium 1 is related to the angle of transmission θ 2 in medium 2 through the relation given in Figure 1.31. If the index of refraction n1, 2 of both media are weakly dependent on the frequency (weakly dispersive media), then the deviation angle θ1 − θ 2 is also weakly dependent on the frequency: the multicolor incident beam (white arrow) is only slightly dispersed in medium 2. If medium 2 is a 2DPC, a super-refraction or super-prism effect may manifest itself, in terms of a strong angular dispersion of the multicolor incident beam impinging onto the photonic crystal. This effect occurs when the “targeted” point of the dispersion characteristics of the 2DPC coincides with an extreme, for example the K point of the high symmetry direction ΓΚ , at the first Brillouin zone boundary, shown by the arrow in Figure 1.31. All that remains to be done is to adjust, on the one hand, the central frequency of the incident beam to that of the K point and, in addition, its direction in such a way as to make the projection of the incident wave-vector along the interface direction (here, the ΓΚ direction of the 2D PC) coincide with that of the ΓΚ vector (which corresponds to the retained direction of propagation in the 2DPC); in the case illustrated in Figure 1.31, this choice leads to ki ≅ ΓΚ . The close 2

proximity of the extreme results in a strong variation rate of the wave-vector in the photonic crystal as a function of frequency, and, therefore, in a strong dispersion of its propagation direction, the module of its projection being set. This phenomenon may be used for an efficient spatial separation of the different wavelength components of the multicolor incoming beam. 1.5.2.2. Strategies for vertical confinement in 2DPC waveguided configurations Two approaches are used to ensure the guiding or vertical confinement of photons. In the so called “substrate” approach, vertical confinement is “weak”, which means that the vertical structuring of the optical index is achieved by a low index contrast between the dielectric cladding and core layers. Typically the core guiding layer is a semiconductor layer (through which the 2DPC is etched), epitaxied onto a semiconductor substrate, with a slightly lower optical index. A semiconductor cladding or barrier layer (with a slightly weaker optical index than the substrate) may be inserted in between for a fine adjustment of the vertical

58

Nanophotonics

electromagnetic distribution. This is the usual configuration for classical integrated optoelectronics based on III–V compound semiconductors. The substrate approach is therefore fully compatible with the classical technology currently in use. It enables us to take advantage of the well controled coupling schemes between the devices and input-output optical fibers, owing to the relatively comfortable thickness (around 1–2µm, for monomode operation in the 1.5µm wavelength range) of the guiding or vertical confinement zone. There are drawbacks to this approach, however: it requires us first to control the fabrication of the holes of the 2DPC with a very large shape ratio; the depth of the holes must indeed exceed significantly the thickness of the guiding zone in such a way as to minimize optical losses in the semiconducting substrate (LAL 01); in addition, the large index of the cladding substrate results in a light line with a rather weak slope (like the inverse of the optical index), which leaves very little room for the pure loss-less waveguided modes, which are not allowed to couple with radiated modes below the light-line. In the so called “membrane” approach, vertical confinement is strong: guiding of light is achieved in a high index semiconductor membrane surrounded by low index cladding or barrier layers (for example an insulator such as silica, or simply air). The virtues and drawbacks of the substrate approach become precisely the drawbacks and virtues of the membrane approach: in monomode operating conditions the membrane is very thin – around a fraction of a µm; the result of this is that low loss coupling schemes with an optical fiber are not easily achievable; however, the positive aspect of this lies in the relaxed technological constraints for the fabrication of the 2DPC (holes with an approximately uniform shape ratio). In addition, one may rely on a “reservoir” of waveguided modes below the light-line that is far more comfortable than with the substrate approach. In addition, strong vertical confinement, leading to a reduced volume of modes, results in a substantial increase in their merit factor. But the essential asset of the membrane approach will be revealed fully in section 1.7, where 2.5D Microphotonics will be briefly discussed: its rationale lies in particular in the “harmony” that is clearly apparent between the lateral structuring (2D PC) and the vertical structuring in terms of optical index contrasts. We will see that this particular feature enables us to contemplate with ease coupling between waveguided modes and radiated modes, which is otherwise considered to be a problem. The membrane approach, in this respect, enables tight control of this coupling and its full exploitation, thus opening the way to a considerable widening of the functionality provided by 2DPC, no more restricted to 2D operations but, instead, freed to the third dimension of space. In the rest of this chapter we will concentrate on the membrane approach, which lends itself to more accessible optical objects from the conceptual point of view. A schematic view of basic building blocks for the membrane approach is shown in Figure 1.32.

Photonic Crystals: From Microphotonics to Nanophotonics

nhigh ~3 nlow =1 Suspended membrane in air

59

nhigh ~3 nlow ~1 to 2 Membrane bonded on a low index layer, silica for example

Figure 1.32. The basic building blocks of the membrane approach

The membrane including the 2DPC may be suspended in air: this is the “ideal” situation from the basic study point of view. For practical applications, where the thermal budget should be considered carefully, bonding of the membrane on a low index substrate (silica for example) is to be applied. In the latter situation, advantage is also taken of improved mechanical stability as well as easier technological conditions (for the electrical contacting of the devices, for example). These two situations will be considered in the following sections. 1.6. 2D photonic crystals: basic building blocks for planar integrated photonics 1.6.1. Fabrication: a planar technological approach An essential advantage of photonic integrated circuits based on 2DPC lies in their fabrication procedure which fits with the planar technological approach, familiar in the world of silicon microelectronics. Without sacrificing their generality, results presented as illustrations in the following sections concern essentially the cases of InP membranes either suspended in air or bonded onto silica on silicon substrate. This latter technological approach opens up promising prospects for heterogeneous integration of optoelectronics devices based on III–V compound semiconductors with silicon microelectronic circuits. The devices are designed for operation wavelengths of InP and related materials, that is in the 1.5µm range. 1.6.1.1. 2DPC formed in an InP membrane suspended in air The heterostructure to be suspended is epitaxied on a semi-insulating InP substrate. It comprises a “guiding” InP layer with half-wavelength optical thickness (around 250nm) and may include active layers for emission or detection (for

60

Nanophotonics

example InAsP quantum wells or InAs based quantum box layers), located at midheight in the InP layer. The InP layer is formed on an InGaAs sacrificial layer, which is eliminated in order to suspend the former, leaving a quarter-wavelength air gap: the elimination of the sacrificial layer is achieved by applying a wet etch surface micromachining procedure. The thicknesses of the sacrificial layer and guiding layer are chosen so as to maximize the coupling of the active layer with fundamental TE waveguided mode (monomode operation) and to inhibit the direct coupling to radiated modes. Heterostructures with quantum box layers provide a variety of interesting features, especially a weak absorption of the guided light together with a wide emission spectral range at room temperature (LET 01): these characteristics allow for the exploration of the modal properties of 2DPC in a wide spectral range (1,250–1,650nm). The photonic crystal is fabricated using electron beam lithography, whose technological steps are described in detail in Pottier et al. 1999. The lattice parameter of the triangular 2DPC is about 500nm, and the filling factor of the holes ranges from 0.35 to 0.5. 1.6.1.2. 2DPC formed in an InP membrane bonded onto silica on silicon by molecular bonding

a : Epitaxy

c : SiO2 - SiO2 bonding III-V barriers (InP) and active layer (InAsP) InGaAs sacrificial layer

SiO2

InP substrate Silicon substrate

Silicon wafer d : substrate removal

b : Oxide deposition

SiO2 E. Jalaguier, S. Pocas, B. Aspar CEA-LETI, Grenoble, France

Full InP wafer bonding

Figure 1.33. Technological steps in the molecular bonding procedure

Photonic Crystals: From Microphotonics to Nanophotonics

Single defect H1 Resonant cavity

61

Hexagonal H5 Resonant cavity

Add-drop selective filter

Specifically designed waveguide

Single line defect waveguide, including resonant cavities

Figure 1.34. SEM micrographs of various integrated microphotonic devices based on 2DPC

The heterostructure is similar to that described above. The InP substrate, including the heterostructure, is bonded by the SiO2-SiO2 molecular bonding procedure (see (MON 01) for a complete description of the technological procedure, developed at CEA-LETI): a schematic view of the successive technological steps is presented in Figure 1.33, which also shows a view of a full wafer bonded InP membrane. The InP substrate is eliminated by selective wet etching (HCl solution). The sacrificial InGaAs layer is finally etched off by selective wet etching (FeCl3 solution). The thickness of the SiO2 layer, below the heterostructure is around 800nm, which is enough to get rid of any significant evanescent coupling of the waveguided modes with the silicon substrate. The fabrication of the 2DPC is then conducted as described previously. Figure 1.34 provides a restricted sampling of the extreme variety of conceivable micro-photonic devices based on 2DPC. In the following section, we explore the principal building blocks on which they are based.

62

Nanophotonics

1.6.2. Localized defect in the PBG or microcavity

H2

f (air %)

Plane wave calculation GES, Montpellier 2D FDTD Simulation Figure 1.35. H2 cavity: magnetic field distribution of cavity modes (2D FDTD determination) and localized states introduced in the PBG

The concept of localized defects or microcavities introduced and analyzed in detail in sections 1.3.5 and 1.3.6 devoted to 1DPC (ideal as well formed in a planar dielectric waveguide), may be extended to the case of 2DPC, as already noted in section 1.5.2. A local break in the periodic lattice constitutes a defect which behaves like a microcavity, which may trap or localize photons in space and produces localized states, allowed within the PBG energy range. The most widely investigated defects or microcavities are obtained by the omission of a certain number of holes in the periodic lattice, such as in the particular case of Hn type hexagonal shaped cavities, n standing for the number of missing rows per side of the hexagon. The case of the H2 cavity is presented in Figure 1.35: the energy levels of the localized states in the PBG (white zone), or cavity modes, are calculated as a function of the hole filling factor (“plane wave” modeling realized at GESMontpellier: (MON 03)); the experimental points are derived from photoluminescence measurements. The magnetic field distribution of two types of cavity modes (“whispering gallery” mode on the left side, transverse mode on the right side) is also shown (FDTD simulation).

Photonic Crystals: From Microphotonics to Nanophotonics

1300

1350

1400

1450

1500

1550

Longueur d'onde (nm) Wavelength (nm)

1600

H2

1250

1300

1350

1400

1450

1500

Longueur d'onde (nm)

Wavelength (nm)

1550

1600

H5

2,0

PL intensity (u.a.) Intensité PL (u.a)

Intensité(u.a.) PL (u.a) PL intensity

PL intensity (u.a.) PL intensity Intensité (u.a.) PL (u.a)

H1

63

1,5

1,0

0,5

0,0

-0,5 1250

1300

1350

1400

1450

1500

1550

1600

Longueur d'onde (nm)

Wavelength (nm)

Figure 1.36. Spectral signatures of hexagonal cavities derived from photoluminescence measurements

Figure 1.36 shows the spectral distribution of the modes of hexagonal cavities with different sizes as derived from photoluminescence measurements: the latter is the manifestation of spontaneous recombination processes of the active medium, which is to a large extent controled by the cavity modes, whose spectral density dominates the total available density of optical modes; the photoluminescence spectrum is therefore a spectral signature of the cavity modes (see (MON 03) for further details). In the same way as in the case of cavities formed in 1DPC (section 1.3.5), we can observe that the number of modes in a given spectral range increases with the size (here the surface) of the cavity. The elementary H1 cavity (one single missing hole) possesses only one mode (yet doubly degenerated) in the investigated spectral range. As explained in section 1.3.6, the spectral widening δω or δλ of the cavity modes is like the inverse of their lifetime τ , which is principally controled by the coupling rate 1/τc with the radiated continuum (if the PBG area around the cavity is large enough to prevent significant lateral escape of photons, which means τ ≅ τ c ). The lifetime of cavity modes varies like their quality factor Q (see equation (6)): the experimental values of Q reported in the literature are in the range of 102–105. The talented designer will have the capacity to reduce the radiation of the cavity inside the light cone, that is to minimize the k vector Fourier components above the lightline (see Figure 1.18) and therefore to produce strongly “resonant” structures with high quality factors. If, in addition, the large quality factor is achieved with a small

64

Nanophotonics

volume cavity (hosting therefore small volume optical modes8), then the merit factor F of the modes, which coincides with the Purcell factor (see also Chapter 4 in this book) may be very large, which is highly desirable for exacerbating the spontaneous recombination rate of an active medium at the resonance frequency and for producing low threshold lazer micro-sources. 1.6.3. Waveguiding structures The operation of classical optical waveguides is based on refraction phenomena: the existence of waveguided modes is made possible by the total internal reflection processes occurring at the boundary between the guiding zone and the external world, thus confining optical modes whose dispersion characteristics are located below the light-line, as presented in section 1.2.4. Guiding structures using photonic crystals are based, in addition, if not exclusively, on diffraction phenomena, which may be exploited to channel photons within an area where the existence of modes is not forbidden.

FDTD simulation Figure 1.37. Magnetic field distribution of guided modes along a linear defect, formed by a missing row of holes in a triangular 2DPC (waveguide W1)

We will concentrate on the case where the guiding structure is formed by a linear defect (one-dimensional structure) introduced in the 2DPC, which operates in the field of PBGs: this configuration has been the matter of a great number of studies reported in the international literature.9 8 Note that the membrane approach is attractive, in that respect, since it allows for a weak vertical extension of the optical modes, as compared to the substrate approach. 9 It is also possible to channel photons in a defect-free 2DPC, in permitted photonic bands, using the so called self-collimating phenomenon: this phenomenon, which is intrinsically diffractive, results from the non-isotropy of dispersion characteristics and from the fact that group velocity, which is oriented toward the propagation direction of energy, is normal to isofrequency curves (see section 1.5.1 and, for example, (CHI 03)).

Photonic Crystals: From Microphotonics to Nanophotonics

65

The canonical example, corresponding to the case of a linear defect simply formed by one missing row of holes ( ΓΚ direction in this example), is shown in Figure 1.37: it is a so called W1 waveguide. The magnetic field distribution of a waveguided mode along the guiding structure is also illustrated: it shows a periodicity which is naturally imposed by that of the 2DPC along the direction of propagation. The waveguided modes are Bloch modes with one dimension (in a three-dimensional world, if the 2DPC is formed in a dielectric membrane): their properties are very similar to those already described in section 1.3.6. Their dispersion characteristics are governed by the coupling processes induced by longitudinal 1D periodic “corrugation”, and exhibit bands where propagation is allowed, separated by PBG (so “mini-stop bands”: (OLI 01)), with photonic band edge extremes.

3D MODELING k vector along the guide direction

Odd modes Odd modes Even modes

Light-cone

Allowed bands of the PC

Figure 1.38. Dispersion characteristics of a W1 waveguide (from GES, Montpellier)

The normalized dispersion characteristics of the W1 waveguide are shown in Figure 1.38. The k propagation vector is oriented along the longitudinal direction of the guide and is limited to the first Brillouin zone. Spatial magnetic field distributions are also shown at various points in the dispersion characteristics: note that the symmetry of the modes depends heavily on the propagation conditions and on the nature of the guiding processes. For example, in the linear region, which corresponds to the fundamental mode and to a spectral range where propagation is “fast” (relatively large group velocity), the “refractive” contribution to waveguiding is rather large, as in classical waveguides (the effective index of the guiding zone, which is free of holes, is larger than that of the 2DPC): the waveguided modes are well confined within the guide. Close to the extremes, on the other hand, where the modes are slowed down, the field distribution is more spread inside the 2DPC, where photons are not allowed to stay: the principal contribution to the guiding and

66

Nanophotonics

confinement of photons within the guide is now taken over by diffractive processes, which are specific to photonic crystals. 1.6.3.1. Propagation losses in a straight waveguide The reader will have noted that dispersion characteristics are partly located above the light-line (within the light cone, light gray area in Figure 1.38), which indicates that guided modes may, in these conditions, couple to the radiated continuum, and naturally result in propagation losses: the waveguided modes are so called lossy modes. This is unlike optical modes confined below the light-line, which may, theoretically, propagate without any loss. It is therefore relevant to design structures where as much room as possible is left to “fast” waveguided modes, with large group velocity, below the light-line. The simple W1 waveguide is not very appropriate, in that respect. A possible improvement is illustrated in Figure 1.39, which shows a modified W1 waveguide: it includes a row of shifted holes, within the core of the guiding zone! This results in a reduction in the effective index of the waveguided modes, leading to an increase in their energy as well as in the slope of the dispersion characteristics below the light-line, which turns to shift the fast fundamental mode below the light-line (GRI 03).

Figure 1.39. Modified W1 waveguide in order to promote fast guiding of photons below the light-line

Constraining photons to the area below the light-line is necessary, but not sufficient in practice to ensure loss-less propagation of light: technological imperfections in the guide (promoting unwanted coupling with the radiated continuum) and the insertion of photons constitute other loss factors, which do not spare optical modes below the light-line. A number of theoretical and experimental works aimed at evaluating and minimizing losses in 2DPC-based waveguides have been reported recently in the literature. Here we will quote an elegant technique of characterization consisting of closing the two ends of guiding sections with variable lengths, thus forming linear cavities, whose resonant modes may be analyzed

Photonic Crystals: From Microphotonics to Nanophotonics

67

spectrally (LET 01): optical losses per unit length of the guide may be derived from the quality factor of the cavity modes; in addition, it is possible to extract the dispersion characteristics of the waveguided modes from the derivation of the free spectral range of the cavity modes and to determine their kinetic properties (group velocity). The best results reported so far (a fraction of dB/mm) are still below the performances accessible to classical optoelectronic waveguides. It turns out, however, that in the prospect of the fabrication of very compact systems, requiring photons to be transported over short distances (not exceeding a few hundred µm), these results may already be considered to be acceptable. It must also be admitted that, unfortunately, these “good” results can be obtained in rather narrow spectral bandwidths (allowed bands for fast propagation below the light-line). 1.6.3.2. Bends

Injected signal

R

1/τ0 1/τ1

ω0 1/τ2

T

Figure 1.40. Conceptual representation of a bend

Transportation of photons necessarily implies the presence of bends to give them a chance to reach their final destination. Let us recall that 2DPC were considered to be very promising, in terms of the production of ultra-compact guides and sharp bends, after the pioneering theoretical work published by the MIT group (JOA 95). A first order argument naturally leads to the idea that PBGs do not have any alternative other than following the corridor that is opened to them when they come to a bend. The reality is different, from both conceptual and practical points of view. A bend should indeed be viewed as a localized defect, resulting in a break in the one-dimensional periodic structure that constitutes the straight waveguide. It must be considered as a microcavity or resonator, which provides a resonant transfer of photons between the two sections of straight waveguides which are attached to it, as illustrated schematically in Figure 1.40. This conceptual representation of a bend is general and may apply to classical guiding structures. In the classical situation, a bend may be considered as a defect

68

Nanophotonics

breaking the continuous (and no longer periodic) translational symmetry of the straight waveguide.

Figure 1.41. FDTD simulation of the magnetic field distribution around a bend

If we use a modal type terminology, the bend can be considered as hosting localized resonant modes, which, through their coupling to the input and output waveguided modes, provide the transfer of the injected signal. The coupled mode theory may be used for a relatively simple analytical description of the structure. Such simulation methods as the FDTD technique allow for a precise modeling of its behavior. Figure 1.41 shows the magnetic field distribution, obtained by 2D FDTD simulation10, provoked by the presence of a 120° bend in a W1 waveguide. One observes that the transmission T of the input signal is strongly limited by the bend. In fact, a number of conditions have to be met in order to achieve a near-uniform transmission ratio. Let us express these conditions using the terminology of coupled mode theory: – The transmission can be achieved only within the spectral bandwidth around the resonant frequencies of the bend, considered as a resonant cavity: this resonance must be therefore the widest possible in the spectral domain, which means that the coupling between the cavity modes (whose lifetime τ must be the smallest possible, for a maximum spectral widening) and the waveguided modes should be very strong (which is equivalent, in Figure 1.41, to minimizing the τ 1 and τ 2 time constants): it is therefore appropriate to achieve the best spectral and spatial (especially in terms of symmetry) overlaps between the cavity modes and the waveguided modes. 1 – The coupling rates should be identical, in order to prevent the occurrence

τ 1, 2

of a finite reflection rate of the injected signal.

10 An ideal 2DPC is considered for this simulation.

Photonic Crystals: From Microphotonics to Nanophotonics

69

1

in Figure 1.41, as a result τ0 of cavity mode coupling with the radiated continuum, should be minimized. If not, the transmission is reduced not only for radiated losses, but also for the reflection. – It is advisable to operate in a monomode waveguided regime, to prevent the risk of unwanted coupling between different order orthogonal waveguided modes, promoted by the localized defect which is formed by the bend and resulting in extra reflection. – Optical losses at the bend, expressed by the ratio

These various conditions may be summarized by the relation below: 1

τ0

≤≤

1

τ

=

2

(27)

τ 1, 2

Classical bends have the definite advantage over PC based bends that they “naturally” comply with and/or circumvent most of the previous conditions. Regarding compactness, which was argued to be a very attractive feature of PCbased bends by the MIT group, classical high index contrast refractive microwire waveguides (silicon wires on silica, for example) are strong contenders to PC waveguides: they allow for very sharp bends (bend radius around one micron), without significant theoretical losses, as a result of the evanescent coupling of waveguided modes with the radiated continuum. May, therefore, PC-based waveguides be considered to be at all useful? 1.6.3.3. The future of PC-based waveguides lies principally in the guiding of light 2DPC-based waveguides are not well suited, in general, to photon transportation or to the optical transfer of information: it is more appropriate to restrict their use to the “smart” treatment of optical signals. It should not be forgotten that a guide formed in a 2DPC is itself a one-dimensional periodic structure, which behaves like a 1DPC: it is therefore advisable to exploit the natural virtues of photonic crystals which allow for tight control of the kinetics of photons, that is for their efficient slowing down and their trapping in a very limited space for a significant amount of time. It is therefore preferable to use 2DPC-based waveguides precisely under the conditions where they are inefficient light carriers, that is at operation points of their dispersion characteristics where their group velocity is limited, especially around extrema. Figure 1.42 shows a schematic representation of the general scenario that should be followed for the use of 2DPC-based waveguides. Compact sections of “slow” PC-based waveguides, which are photon storing zones where they will stay during

70

Nanophotonics

the time required for the optical signal process, are connected by fast, classical refractive waveguides. The tapering transition zone between fast and slow sections is essential in the design of the whole structure: it is meant to minimize insertion losses, which manifest themselves by unwanted reflections and coupling to the radiation continuum.

Fast transport refractive section

Long stay / confinement diffractive section

Adaptation (taper for example)

Figure 1.42. General scenario for the use of a 2DPC-based waveguide

This general scenario is expected in result in the production of a wide range of compact devices with very diverse functionality: let us mention, for example, electroor thermo-optical modulators based on Mach-Zhender interferometers, when full use is made of the strong phase optical index dispersion or of the large group optical index, which occurs near an extremum of the dispersion characteristics; or efficient, although compact, chromatic dispersion compensation structures; or devices making the best use of non linear phenomena, in terms of the required input optical energy, owing to strong photon confinement. The list could go on. 1.6.4. Wavelength selective transfer between two waveguides Wavelength Division Multiplexing is today considered to be a powerful enabler for the optical transfer and treatment of information, particularly for telecommunications applications. This approach is also meant to relax high density scale integration constraints in microelectronics by opening the way to the so called Systems On Chip, combining microelectronic and microphotonic integrated circuits. In this respect, the multiplexing-demultiplexing function, which consists of adding to or dropping from a waveguide selected wavelengths, is an essential element of optical signal processing. The principle underlying this function is presented schematically in Figure 1.43. It consists of the use of a mediator between two waveguides, which is meant to

Photonic Crystals: From Microphotonics to Nanophotonics

71

promote the transfer of certain wavelengths from one to the other. The selected wavelengths are set by the intersections between the dispersion characteristics of the two guides with that of the mediating section. Because the waveguides are meant to provide “fast” transportation of photons, the mediator should have rather flat dispersion characteristics, in such a way that their intersections with those of the fast waveguides can be well defined; this means that it should operate in the regime of slow Bloch modes, or even trapped modes (in extreme cases).

Guides

ω

λ0

Mediator Mediator Σλ− λ0

Σλ

K

λ selection

Figure 1.43. General coupling principle between two waveguides with wavelength selectivity

One can therefore contemplate the potential importance of PC-based structures for that purpose. Wavelength selectivity δλ is essentially related to the lifetime or stay duration τ of optical modes in the mediator:

δλ proportional to

1

τ

=

1

τ0

+

(28)

1

τc

τ c is the coupling time constant of the mediator modes with fast waveguided modes and τ 0 is the lifetime of modes related to optical losses (for example controled by coupling with the radiation continuum and/or by absorption). ω

Guides

Cavity Injected signal

λ0 k//

Figure 1.44. A monomode microcavity cannot provide a directional transfer of wavelengths between two waveguides

72

Nanophotonics

The direction of the wavelengths’ selective coupling is another important characteristic of the add and drop function: it depends essentially upon the symmetry of the mediator modes, which take care of the wavelength transfer. In the simple canonical case where, for example, the mediator is formed by a monomode cavity (in the considered spectral range), the transfer cannot be directional, as illustrated in Figure 1.44. In steady state regime, the cavity mode which is generated by the coupling with the waveguided mode injected in the input waveguide is in turn coupled to both waveguides and results in counter-propagating waveguided components, as dictated by the symmetry of the structure and as allowed for by the dispersion “characteristics” of the cavity: the latter is indeed restricted to a monochromatic horizontal line, where all k vectors are authorized. Consequently, the input signal is partly reflected in the input waveguide, and partly transmitted in both directions of the output waveguide11. The MIT group (FAN 98) proposed a solution to directional transfer based on the use of two degenerated cavity modes, whose symmetry properties are judiciously chosen, with the further condition that the coupling rates of each of these modes with the waveguides modes are equally balanced. These conditions are naturally met in the well known classical configuration where two refractive waveguides are resonantly coupled via a micro-disk or micro-ring type cavity (provided that the diameter is larger than the operation wavelength). Meeting these conditions with PC-based waveguides and cavities is tricky from a technological point of view, the size resolution being very high (to the order of one nanometer). Another solution based on photonic crystals, with fewer technological constraints, consists of the use of a mediator formed by a section of “slow” waveguides, which operates around an extreme of its dispersion characteristics (HAT 05). The resonant mode of the mediator is not localized as in a microcavity, but it is a “propagating” slow Bloch mode: for this mode to retain its propagating character, its lateral extension must remain inferior to the size of the slow waveguide during its time in the mediator; it would otherwise behave like a microcavity mode, and the directionality would be lost. The slow PC-based waveguide, used around an extreme of the dispersion characteristics, enables this last constraint to be overcome, under the best compactness conditions (see sections 1.3.3 and 1.3.4, and equation

11 It can be shown easily, using coupled mode theory, that in the absence of an output waveguide, the coupling of the input waveguide with the monomode cavity results in the total reflection of the input signal, if optical losses are negligible. Asano et al. 2003 have used this configuration to extract wavelengths from a guide by using radiation optical losses of the cavity in free space; although attractive, this approach is not expected to result in directional operation (50% at most of the input signal is extracted at resonance, the remainder being either reflected or transmitted in the input waveguide).

Photonic Crystals: From Microphotonics to Nanophotonics

73

(14)). The operation of this type of wavelength selective and directional coupling is illustrated in Figure 1.45, with a specific example. ω

Normalized response

k//

S4

S3

S1

S2

Figure 1.45. Wavelength selective and directional coupling between two waveguides: use is made of a slow Bloch mode “resonator”. The spectral response and the magnetic field distribution (2D FDTD simulation) corresponding to the specifically chosen example are also shown

1.6.5. Micro-lazers The operation of a lazer is based on the interaction between an active light emitting medium and the optical modes of a photonic structure which will have been designed to promote their mutual coupling: the objective is to reach a threshold density of photons in the active medium, for the highly coherent so called stimulated emission to overcome the poorly coherent spontaneous emission in the active medium. The general approach is to design the photonic structure in such a way as to confine optical modes within the space occupied by the active medium. The reader will have realized that the intrinsic qualities of photonic crystals, analyzed in previous sections (see particularly sections 1.3.4 and 1.5.2), make them ideal for achieving the function of photon confinement in a very compact way. Let us recall that, in general, the lateral confinement of photons can be achieved in a 2DPC, either by trapping them in a localized defect or microcavity giving rise to a localized mode within a photonic band gap, or by slowing them down in a slow Bloch mode at an extreme of the dispersion characteristics. Those two approaches result in two classes of lazers, microcavity lazers and Bloch mode lazers. Figure 1.46 shows the different types of lazers, from the point of view of their operation on dispersion characteristics.

74

Nanophotonics

Surface emitting Bloch mode lazer: band edge de-localized modes

ω

PBG

k

Micro-cavity lazer: modes localized in the PBG In plane emitting Bloch mode lazer: band edge de-localized modes

Figure 1.46. Different types of micro-lazers, as a function of their operation on dispersion characteristics

In the Bloch mode lazer class12, two types of devices can be distinguished: – Lazers designed for in-plane emission: the operation point coincides with an extreme located below the light-line (the K point in the example in Figure 1.46; the emission is preferentially oriented toward the ΓΚ directions in the plane). – Lazers designed for surface emission, that is to say in free space: the operation point coincides with an extreme located above the light-line (the Γ point in the example of Figure 1.46), resulting in vertical emission. The operation principle of this type of lazer implies therefore that waveguided Bloch modes may couple to the radiated continuum; the coupling rate must however not be so strong as to “destroy” the Bloch mode resonance, that is to say, during its lifetime or its time of interaction with the active medium it must not be reduced to such an extent that the stimulated emission threshold cannot be reached. It turns out that, for reasons of symmetry, coupling with the radiated continuum may theoretically be forbidden at the Γ point, for 2DPC with specific symmetry (triangular lattice, for example): this property ensures an efficient confinement of the waveguided mode within the membrane waveguide, whereas emission close to the vertical direction is permitted. 1.6.5.1. Threshold power The threshold power of the lazer is the minimum power required for the stimulated emission to exactly compensate for all photon loss processes (absorption, optical losses).

12 So called DFB classical lasers are also Bloch mode lasers; the novelty of 2DPC lies first in their compactness, as explained at length in this chapter, and, second, on the extra dimension that they offer.

Photonic Crystals: From Microphotonics to Nanophotonics

75

For microcavity lazers, the threshold power is proportional to the volume of the cavity and is a decreasing function of the lifetime of the cavity mode resulting in the lazer effect (LET 05). The strong lateral confinement of photons which are trapped in the localized defect within the PBG of the 2DPC, combined with the thin vertical confinement provided by the membrane approach, enables the production of microcavities whose volume does not exceed a fraction of µm3 (for an operation wavelength around 1,5µm); this opens the way for the production of very low threshold power lazers, as far as the lifetime of the cavity mode can be kept long enough, or the various sources of losses can be minimized – especially the optical losses of the cavity. For Bloch mode lazers which operate at a band edge extreme, the threshold power is proportional to the curvature α of the dispersion characteristics at the extreme (LET 05). We see, once more, that an essential quality of photonic crystals is to offer dispersion characteristics with very low curvature band edge extremes. That the threshold power is proportional to α is a direct consequence of the fact that the lateral optical mode confinement is also proportional to α . Let us recall that the surface extension of a Bloch mode at an extreme can be written S BM ≈ ατ , where τ is its lifetime (see section 1.5.2, footnote 4). 1.6.5.2. Example: the case of the surface emitting Bloch mode lazer Experiments with the three types of micro-lazers described above have been reported in the international literature, following the pioneering work of the group at Caltech published in 1999, which concerned the production of the first microcavity lazer (H1 type) formed in an InP suspended membrane (PAI 99). Other published works may be pointed out, such as Hwang et al. 2000 on microcavity lazers and Ryu et al. 2002 on Bloch mode lazers. The group at the Ecole Centrale de Lyon has experimented with the three types of lazers, formed in InP membranes bonded onto silica on silicon substrate (MON 01, 02; MOU 03), following the technological procedure described in section 1.6.1. Bonding of the membrane onto silica improves the thermal budget considerably as compared to the configuration where the membrane is suspended in air. We present below, as an illustrative example, the case of surface emitting Bloch mode lazer (MOU 03). The photonic crystal consists of a graphite lattice (Figure 1.47), which can be viewed as an array of H1 coupled cavities, formed in a triangular lattice.

76

Nanophotonics

Graphite crystal

Exploitation of flat bands in Γ (k//=0) (above the “light-line”)

Figure 1.47. Band of a surface emitting Bloch mode lazer formed in a graphite type 2DPC

This particular 2DPC exhibits band edge extremes at the Γ point with very low curvature (Figure 1.47). One of these extremes is shown in the example below.

Figure 1.48. Emission spectra of the surface emitting lazer formed in a graphite lattice 2DPC, for different hole filling factors (f). The plot of the emitted power versus the pumping power indicates a threshold power of 40 µW for f =19%

Emission spectra of the lazer are shown in Figure 1.48 for different hole filling factors f, as well as the spontaneous emission spectrum of the non-structured membrane. The device is optically pumped in a quasi-steady state regime and operates at room temperature. The peak intensity for the optimum filling factor (f = 19%) is larger than the spontaneous emission power by 5 orders of magnitude. To increase f, as might be expected, the emission peak is blue shifted as a result of a reduction in the effective optical index of the membrane; at the same time, the

Photonic Crystals: From Microphotonics to Nanophotonics

77

emission yield drops rapidly due to a decrease in the modal gain (not shown in the figure). The effective threshold pumping power, for the optimized device, is very weak and does not exceed 40µW. The pumped area where the simulated emission process takes place is very limited and does not exceed 2 to 4µm in diameter: this is a clear demonstration of the outstanding ability of 2DPC to confine laterally slow Bloch modes. 1.6.6. Epilogue Our trip into the world of 2D photonic crystals has come to an end; we should however be convinced that we have tasted some of the wide variety of landscapes that 2DPC has to offer, albeit restricted to a two-dimensional universe. The last example presented in the previous section (surface emitting lazer) opens the door to an even wider range of opportunities which may be accessible when freeing 2DPC towards the third dimension. We present these new emerging developments briefly in the last part of this chapter. 1.7. Towards 2.5-dimensional Microphotonics 1.7.1. Basic concepts Photonic devices based on 2DPC are principally aimed at forming the basic building blocks of integrated photonics and are designed for in-plane waveguided operation. We recall that these devices are very attractive from the point of view of fabrication, the technological schemes to be adopted being compatible with planar technological approaches. We recall also that the operation of photonic integrated circuits based on 2DPC may be deeply affected by optical losses resulting from unwanted diffractive coupling of waveguided modes with the radiation continuum. This problem of optical losses, which is considered as hindering the operation of 2D photonic integrated circuits based on 2DPC, can be approached from a completely different perspective: instead of attempting to confine the light entirely within waveguide structures, 2D structures can be deliberately opened to the third space dimension by controling coupling between waveguided and radiation modes. In this approach, exploitation of the optical power is achieved by accurately tailoring optical radiation into free space. The surface emitting micro-lazer (section 1.6.5) is an example of this approach: coupling between waveguided modes and radiated modes is authorized, but its rate is controled accurately, allowing for vertical emission while retaining the strength of resonance and, therefore, achieving weak threshold power. This is a simple and

78

Nanophotonics

convincing illustration of a planar technological approach resulting in a photonic device freed from the bidimensional universe.

(ω, k// )

ω BI

k//

k

Figure 1.49. Illustration of the resonant coupling between a waveguided mode and a radiated mode

A major extension of planar technology has recently been proposed, through exploitation of the third (“vertical”) dimension by using a so called multi-layer approach, where lateral high index contrast patterning of layers would be combined with vertical 1D high index contrast patterning: here it is more appropriate to think in terms of “2.5 dimensional” photonic structures, in which there is interplay between waveguided-confined photons and radiated photons propagating through the planar multilayer structure (LET 03). A simple illustration of this approach is the use of a plain photonic crystal membrane as a wavelength selective transmitter/reflector: when a light is shone on this photonic structure, in an out-of-plane (normal or oblique) direction, resonances in the reflectivity spectrum can be observed. These resonances, so called Fano resonances (AST 99), arise from the coupling of external radiation to the guided modes in the structures, whenever there is a good match between the in-plane component of the wave-vector of the incident wave and the wave-vector of the guided modes (see Figure 1.49). If the lateral size of the illuminated membrane is infinite, the spectral width of the resonance is like the inverse of its lifetime τ , that is the lifetime of the waveguided mode, with τ = τ c , where τ c is simply the coupling time constant between waveguided and radiated

Photonic Crystals: From Microphotonics to Nanophotonics

79

plane-wave modes13. In real devices, the lateral size of the illuminated area is limited, and the lifetime of the resonance is also controled by the lateral escape rate 1 of the waveguided mode out of this area; this escape rate should be considered τg

as a loss mechanism for devices which are designed and intended to operate “vertically”. In these real conditions the lifetime of the resonance is written as: 1

τ

=

1

τc

+

1

τg

≈ δω ,

(29)

where δω is the spectral widening of the resonance. The ability of high index contrast PC to slow down photons and to confine them laterally, especially at the high symmetry points (or extrema) of the dispersion characteristics, as explained a number of times earlier in the chapter, allows for very good control over the lateral escape losses and results in very compact devices. If we now consider a multilayer structure, the strong vertical 1D modulation of the optical index allows for a fine and efficient “carving” of the density and vertical field distribution of radiated modes, using a limited number of layers. In summary, 2.5D Microphotonics, combining lateral 2DPC and vertical 1DPC, should provide a very good control over the electromagnetic environment, that is over the distribution of optical modes in 3D real space and time, at a much lower cost than the full 3D approach in terms of technological feasibility: the technological schemes to be adopted are compatible with technological approaches which are normally describable as planar. This multi-layered or multi-level approach is familiar in the world of silicon microelectronics, when it comes, for example, to fabricating multiple levels of electrical interconnections; its usefulness in Microphotonics extends far beyond, from the viewpoint of a considerable extension of accessible functionality.

13 Various factors contribute to the control of the coupling time constant, such as the strength of the periodic corrugation, and the symmetry of the waveguided mode.

80

Nanophotonics

1.7.2. Applications

N I P

Substrate

Figure 1.50. 2.5D photonic structure including several InP membranes suspended in air, with a 2DPC formed in the top membrane (SEM view)

It is clear that the exploitation of the third dimension should widen the domain of Integrated Photonics considerably. In addition to the convincing experiments with surface emitting micro-lazers mentioned above, other outstanding results have recently been published: they concern, in particular, the use of the non-linear response of a 2DPC, enabling the manipulation of Fano resonances and opening the way for new classes of compact surface addressable devices, for full optical routing and signal regeneration (RAI 03). Other areas of Photonics should take advantage of the 2.5D Microphotonics approach. For example, the introduction of 2DPC into MOEMS (Micro Opto Electro Mechanical) devices shows great promise in terms of widening the spectrum of (electro-mechanically actuable) optical functions, achievable with further enhanced compactness structures. Figure 1.50 gives an example of such a MOEMS 2.5D structure. These new types of photonic structures should be applied in various domains, including Optical Telecommunications (tunable or switchable wavelength selective devices, taking advantage of the extra angular resolution provided by the 2DPC), as well as the field of optical sensors: it should be noted that the sensing function should be greatly enhanced, due to the fact that the resonant “portions” of the device, i.e. those where the electromagnetic field intensity is maximum, can be very close to (or even at) the surface of the device. This is in favor of an improved sensitivity and of an easy exploitation of so called functionalized surface sensors The “2.5D Microphotonics” approach also brings about a new opportunity, practicable in terms of technological constraints and rich in terms of extra degrees of liberty, to control the electromagnetic environment at the wavelength scale. It is

Photonic Crystals: From Microphotonics to Nanophotonics

81

therefore the appropriate route for “physicists” looking for ad hoc configurations in Quantum Electrodynamics studies (coupling of active material with the electromagnetic field, threshold-less lazer, directional single photon sources, combination with near optical field studies, etc.). Let us mention, finally, the great potential of 2.5D Microphotonics in biological applications, in the development of “biophotonic chips”, which would be very efficient in terms of “photonic yield”: for example, the accurate control of resonant processes at the surface of photonic microstructures should result in a considerable increase in the collection efficiency of the luminescent signal emitted by fluorescent markers used in biological chips. 1.8. General conclusion The flow of innovations since the late 1980s created by the introduction of the concept of the photonic crystal (YAB 87; JOH 87) is still rather slow, but will no doubt accelerate in the future to an extent which is beyond our full consciousness: it was simply proposed to extend the field of optics to the three dimensions of space, which was rather confined, yet with very successful outcomes, to the one dimensional world of multilayer optical structures. It is now established that the emergence of 3D Microphotonics based on full 3DPC will be significantly delayed, as a result of technological constraints. We hope that the reader will have been convinced that, on the other hand, 2DPC are fully engaged in the process of innovation and that we are experiencing, in that respect, a truly microphotonic revolution. We have shown that 2DPC are very promising for 2D microphotonic integration; there is however a lot left to be done before 2DPC devices are fully integrated in the world of optoelectronics, especially in connection with the solutions required for the control of radiation optical losses. As for so called 2.5D Microphotonics, where 2DPC are deliberately opened up to the third dimension of space, we have seen convincing demonstrations of their ability to generate, in the short term, a wide range of photonic devices (“killer applications”) combining compactness, spatial (angular) and spectral resolution, and whose fabrication meets the standards of planar technology, familiar to the world of microelectronics. It appears that the rising trajectory of photonic crystals will not be inhibited in the long run, provided that appropriate tools are made available for their evolution. In that respect, bottlenecks are still to be eliminated and important R&D will have to be deployed for that purpose: this is true for the modeling and design aspects (especially 3D), whose fast and efficient tools are yet to be built; the technological constraints, dictated by the necessity to control the size of the devices at the nanometer scale, are far from solved. From the latter point of view, it can be stated that we really have entered the Nanophotonic era.

82

Nanophotonics

1.9. References [ASA 03] Asano T, Song B-S, Tanaka Y and Noda S (2003) Investigation of channeladd/drop-filtering device using acceptor-type point defects in a two-dimensional photonic crystal slab. Appl. Phys. Lett. 83 (3): 407. [AST 99] Astratov VN, Whittaker DM, Culshaw LS, Stevenson RM, Skolnick MS, Krauss TF and De La Rue RM (1999) Photonic Band Structure Effects in the Reflectivity of Periodically Patterned Waveguides. Phys. Rev. B60 (24): R16255. [CHI 03] Chigrin DN, Enoch S, Sotomayor Torres CM and Tayeb G (2003) Self-guiding in two-dimensional photonic crystals. Optics Express 11(10): 1203. [CUI 99] Cuisin C, Chen Y, Decanini D, Chelnokov A, Carcenac F, Madouri A, Lourtioz JM and Launois H (1999) Fabrication of three-dimensional microstructures by high resolution x-ray lithography. J. Vac. Sci. Technol. B 17(6): 3444. [GRA 00] Gralak B, Enoch S and Tayeb G (2000) Anomalous refractive properties of photonic crystals. J. Opt. Soc. Am. A17(6): 1012. [GRI 03] Grillet C (2003) Micro-composants optiques à base de cristaux photoniques 2D pour l’optique intégrée. Thesis, Ecole Centrale de Lyon. [FAN 98] Fan S, Villeneuve PR, Joannopoulos JD and Hauss HA (1998) Channel Drop Tunneling through Localized States. Phys. Rev. Lett. 80 (5): 960. [GRU 96] Grüning U, Lehmann V, Ottow S and Busch K (1996) Macro-porous silicon with a complete two-dimensional photonic band-gap centered at 50 µm. Appl. Phys. Lett. 68 (6): 747. [HAT 05] Hattori HT, Grillet C, Letartre X, Rojo-Romeo R, Seassal C and Viktorovitch P (2005) Directional channel-drop filter based on a slow Bloch mode photonic crystal waveguide section. Optics Express 13: 3037. [HWA 00] Hwang J-K, Ryu H-Y, Song D-S, Han I-Y, Song H-W, Park H-K and Lee Y-H (2000) Room-temperature triangular-lattice two-dimensional photonic band gap lasers operating at 1.54µm. Appl. Phys. Lett. 76: 2982. [JOA 95] Joannopoulos JD, Meade RD and Winn JN (1995) Photonic Crystal. Molding the Flow of Light. Princeton University Press, New Jersey. [JOH 87] John S (1987) Strong localisation of photons in certain distordered superlattices. Phys. Rev. Lett. 58: 2486. [KOG 72] Kogelnick H and Shank CV (1972) Coupled wave theory of distributed feedback lasers. J. Appl. Phys. 43: 2328. [LAL 01] Lalanne P and Benisty H (2001) Out of plane losses of two dimensional photonic crystal waveguides: electromagnetic analysis. J. Appl. Phys. 89(2): 1512. [LET 01] Letartre X, Seassal C, Grillet C, Rojo-Romeo P, Viktorovitch P, Le Vassor D’yerville M, Cassagne D and Jouanin C (2001) Group velocity and propagation losses measurement in a single line photonic crystal waveguide on InP membranes. Appl. Phys. Lett. 79: 2312.

Photonic Crystals: From Microphotonics to Nanophotonics

83

[LET 03] Letartre X, Mouette J, Seassal C, Rojo-Romeo P, Leclercq J-L and Viktorovitch P (2003) Switching devices with spatial and spectral resolution combining photonic crystal and MOEMS structures. J. Lightwave Technol. 21(7): 1691. [LET 05] Letartre X, Monat C, Seassal C and Viktorovitch P (2005) An analytical modeling and an experimental investigation of 2D photonic crystal Micro-lasers: defect state (microcavity) versus band edge state (distributed feed-back) structures. Journal of the Optical Society of America B (JOSA B), in press. [LOU 05] Lourtioz JM, Benisty H, Berger V, Gérard JM, Maystre D and Chelnokov A (2005) Photonic Crystals: Towards Nanoscale Photonic Devices, Springer, Berlin-HeidelbergNew York. [MAC 96] Macleod HA (1986) Thin Film Optical Filters. 2nd edition, Adam Hillger Ltd, New York. [MON 01] Monat C, Seassal C, Letartre X, Viktorovitch P, Regreny P, Gendry M, RojoRomeo P, Hollinger G, Jalaguier E, Pocas S and Aspar B (2001) InP 2D photonic crystal microlasers on silicon wafer: room temperature operation at 1.55µm. Electron. Lett. 37: 764. [MON 02] Monat C, Seassal C, Letartre X, Regreny P, Rojo-Romeo P, Viktorovitch P, Le Vassor D’yerville M, Cassagne D, Albert JP, Jalaguier E, Pocas S and Aspar B (2002) InP based 2D photonic crystal on silicon: in-plane Bloch mode laser. Appl. Phys. Lett. 81: 5102. [MON 03] Monat C, Seassal C, Letartre X, Regreny P, Gendry M, Rojo-Romeo P, Viktorovitch P, Le Vassor D’yerville M, Cassagne D, Albert JP, Jalaguier E, Pocas S and Aspar B (2003) Two dimensional hexagonal-shaped microcavities formed in a twodimensional photonic crystal on InP membrane. J. Appl. Phys. 93: 23. [MOU 03] Mouette J, Seassal C, Letartre X, Rojo-Romeo P, Leclercq JL, Regreny P, Viktorovitch P, Jalaguier E, Perreau P and Moriceau H (2003) Very low threshold vertical emitting laser operation in InP graphite photonic crystal slab on silicon. Electron. Lett. 39: 526. [OLI 01] Olivier S, Rattier M, Benisty H, Weisbuch C, Smith CJM, De La Rue RM, Krauss TF, Oesterle U and Houdre R (2001) Mini-stop bands of a one-dimensional system: the channel waveguide in a two-dimensional photonic crystal. Phys. Rev. B 63: 3311. [PAI 99] Painter O, Lee RK, Scherer A, Yariv A, O’Brien JD, Dapkus PD and Kim L (1999) Two-dimensional photonic band-gap defect mode laser. Science 284: 1819. [POT 99] Pottier P, Seassal C, Letartre X, Leclercq JL, Viktorovitch P, Cassagne D and Jouanin C (1999) Triangular and Hexagonal High Q-Factor 2D Photonic Bandgap Cavities on III-V Suspended Membranes. J. Lightwave Technol. 17: 2058. [RAI 03] Raineri F, Cojocaru C, Raj R, Monnier P, Seassal C, Letartre X, Viktorovitch P and Levenson A (2003) Nonlinear Optical Manipulation of Fano Resonances in 2D photonic crystal Slabs. Post Deadline CLEO 2003.

84

Nanophotonics

[ROW 99] Rowson S, Chelnokov A and Lourtioz JM (1999) Two-dimensional photonic crystals in macro-porous silicon: from mid-infrared to telecommunication wavelengths (1.3-1.5 µm). J. Lightwave Technol. 33: 1989. [RYU 02] Ryu H-Y, Kwon S-H, Lee Y-J, Lee Y-H and Kim J-S (2002) Very-low threshold photonic band-edge lasers from free-standing triangular photonic crystal slabs. Appl. Phys. Lett. 80 (19): 3476. [SPI 98] Spisser A, Ledantec R, Seassal C, Leclercq JL, Benyattou T, Rondi D, Blondeau R, Guillot G and Viktorovitch P (1998) Highly selective and widely tunable 1.55 µm InP/Air-Gap micromachined Fabry-Perot filter for optical communications. IEEE Photon. Technol. Lett. 10: 1259. [TAM 88] Tamir T (1988) Guided Wave Optoelectronics, Springer Verlag, Berlin. Chapters 2 and 6. [VAS 02] Le Vassor D’Yerville M (2002) Modélisation de cristaux photoniques bidimensionnels de hauteur finie. Thesis, Montpellier II University. [YAB 87] Yablonovitch E (1987) Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 63: 2059. [YAB 91] Yablonovitch E, Gmitter TJ and Leung KM (1991) Photonic bandgap structure: the face-centered-cubic case employing nonspherical atoms. Phys. Rev. Lett. 67: 2295.

Chapter 2

Bidimensional Photonic Crystals for Photonic Integrated Circuits

2.1. Introduction Within actual Photonic Integrated Circuits (PICs), light confinement is provided by the index contrast that exists between different materials and according to the geometry of a structure. This index contrast mechanism limits the desirable PIC size reduction (performances are then degraded) and the integration of optical functions. Within the periodic environment of a Photonic Crystal (PhC), light confinement can be obtained based on an intentional defect included in the PhC matrix. Devices implementing this multiple Bragg reflection mechanism may overcome these limits [LOU 05]. This chapter is an attempt to demonstrate that passive as well as active optical functions required in Telecoms networks can be fulfilled using PhC-based devices. Passive functions such as guiding, coupling and filtering as well as active functions such as emission and amplification are investigated here. These examples demonstrate that compactness and monolithic integration can actually be reached with PhC-based devices. For the 1.3–1.5µm Telecom wavelength domain, as emission is also addressed, all the structures reported here are fabricated on InP-based materials.

Chapter written by Anne TALNEAU.

86

Nanophotonics

This chapter is divided into three sections: – section 2.2 introduces the concepts that underlie device design, and explains the various performances that are expected of applications. For more in-depth exploration of the concepts, the reader is referred to Chapter 1; – section 2.3 presents the technology required for the fabrication of 2D-PhC structures on InP substrate; – section 2.4, the largest section, details the characterization of passive and active operating devices. Attention is paid to spectrally resolved modal behavior, as well as quantitative assessment of optical performances. 2.2. The three dimensions in space: planar waveguide perforated by a photonic crystal on InP substrate 2.2.1. Vertical confinement: a planar waveguide on substrate For active optoelectronic functions (lasing, light amplification, fast tuning), buffer layers for carrier injection are required on both sides of the active layer. The generic vertical stack on InP substrate which is used for all the devices presented here is shown in Figure 2.1. The guided mode is vertically confined mainly in the high index layer, but spreads partly in the confinement layers, and also in the substrate. This vertical mode confinement results from the “classical” index contrast. In the case of a GaInAsP material layer for 1.55µm emission on InP substrate, the index contrast between the guiding layer and the buffer layers is reduced (∆n/n only 6%), leading to a large vertical extension of the guided mode. Such large spreading has two major consequences: – In order to intercept all the guided mode, holes have to be as deep as its vertical extension: at least 3µm. Such deep etching technology is difficult for 200nm diameter holes (for a PhC with an air filling factor of 35%), in the case of InP-based materials. – The PhC gap lies above the substrate light line (Figure 2.1b), so that any Bloch mode propagating in a defect which is located within the gap can radiate in the substrate; such a mode is known as a leaky mode.

Bidimensional Photonic Crystals

87

Figure 2.1. A: Planar waveguide perforated with the array of holes; B: Dispersion curve for the 2D photonic crystal calculated by the Plane Wave Expansion Method

But this large vertical deconfinement allows, as compared to the membrane case, higher coupling efficiency when light is injected from “classical” structures. This advantage is limited, as the in-plane confinement is still very strong in absolute terms. Quantitative performances and optical losses will be addressed systematically for any device, as a new device is relevant only if its power budget is reasonably mastered. 2.2.2. In-plane confinement: intentional defects within the gap Let us now consider the high index heterostructure in which the PhC is fabricated. Here we investigate an array of holes rather than pillars, in order to generate a connected surface which will enable a metallic contact to be deposited for carrier injection. The triangular lattice gives rise to the widest gap simultaneously for both symmetry directions within the plane. We denote by ΓK the direction of the smaller period in the direct space, and by ΓM the direction of the larger period (see Chapter 1). The fabricated air filling factor is around 35%, leading to a photonic gap only for the TE polarization; TE polarization corresponds to the electric field of the optical mode being parallel to the layers’ plane. Intentional defects introduced within such a PhC matrix can support localized modes [JOA 95]. We consider here defects generated by missing holes. In such an arrangement, the region without holes has a dielectric constant larger than the PhC region, so we have to remember that for any design including this category of “defect” in the PhC, the multiple Bragg mechanism and the index contrast mechanism simultaneously confine the mode.

88

Nanophotonics

2.2.2.1. Localized defects Removing a few holes while following the PhC array symmetry leads to hexagonal cavities. We denote by H1 the smallest cavity obtained by removing a single hole, as each of its sides is one period long. This H1 cavity supports a pair of degenerate modes which is efficiently included in an out-of-plane filter [AKA 03]. A larger cavity, for example the H2 cavity shown in Figure 2.2a, supports a larger number of modes. Cavity eigenmodes are calculated by the Plane Wave Expansion method ([BEN 99], and Chapter 1). We plot in Figure 2.2a the H field component amplitude for two consecutive eigenmodes of the H2 cavity; the mode on the right has its H field component confined on the boundaries, looking like a whispering gallery mode. Such a mode will be of interest for cavity-guide in-plane coupling (section 2.4.1).

Figure 2.2. A: Schematics of an H2 cavity. Absolute value of the H field amplitude for two consecutive eigenmodes; B: Dispersion curves for W1 and W3 guides (ff=40%)

2.2.2.2. Linear defects Removing rows of holes or changing their size creates linear defects that allow Bloch modes to propagate within the gap. All the guides we will consider here are ΓK oriented, along the dense period a. They correspond to an integer number of missing rows, and are denoted by Wi, for i missing rows. Previous characterizations of PhC guides have demonstrated that ΓM oriented guides are more lossy [OLI 02b]. Figure 2.2b displays the dispersion relation for W1 and W3 PhC guides. The dispersion relation plotted here corresponds to the normalized frequency u=a/λ,

Bidimensional Photonic Crystals

89

versus k//, the in-plane k vector component along the symmetry direction chosen, here ΓK. The grey areas are the continuum bands. Even modes are plotted as solid lines, odd modes as dashed lines. These dispersion curves are in this case calculated using FDTD (Finite Difference Time Domain), which basically consists of a full calculation of the field components when discretizing Maxwell equations in both time and space [AGI 03]. Here we use a 2D calculation; the vertical stack is taken into account through an effective index [QIU 02a]. The W1 guide is monomode within the two wavelength domains indicated by braces below and above the odd mode. For light propagation, we will operate the W1 guide within these two domains, as it is always better to operate with a monomode guide to prevent mode mixing. This even mode exhibits a marked cutoff at the Brillouin zone edge, at k// =π/a. The frequency at which this cut-off occurs is strongly dependent upon the air filling factor: thus, measuring the spectral position of this cut-off leads to an optical measurement of the filling factor. The W3 guide is multi-mode throughout the gap. The periodicity that exists along the waveguide boundaries enables, through a contra-directional mechanism, the coupling of two modes of the same waveguide, provided that they have the same symmetry. This coupling mechanism opens a mini-gap, which is shown here by braces in the case of the two first even modes. These mini stopbands are a specific signature of propagation in a PhC guide [OLI 01]. We will see in section 2.4.1 that the spectral width of this mini stopband enables us to estimate the quality of the etched holes. 2.2.3. Losses As has already been pointed out, all the modes supported by defects in the photonic gap are above the substrate light line, so out-of-plane losses occur through coupling to radiation modes. The amount of loss can be calculated using 3D FDTD, but this calculation is very time consuming and hardly takes into account fabrication imperfections such as non-cylindrical holes and etched surface roughness. Benisty et al. proposed describing losses as a fictitious dissipation due to the air contained within the holes, thus adding an imaginary part to the dielectric constant of holes air [BEN 00]. Simulations are then performed in 2D, making their duration reasonable, and include an ε”. This ε” parameter is a phenomenological parameter, which is adjusted when comparing the simulation results to the measured performances. For example, the mini stopband depth and width of the W3 fundamental mode are directly related to losses [QIU 02b]. This ε” parameter enables us to globally qualify the etching technology: finite hole depth, non perfect geometry, roughness (see section 2.4.1).

90

Nanophotonics

2.3. Technology for drilling holes on InP-based materials Characteristic dimensions of holes required for a photonic crystal device operating in the optical domain are well below the micrometer: this technological challenge has delayed the application of PhC in the optical domain. For InP-based materials having an optical index larger than 3 and operating at λ=1.55µm, the gap lies at reduced frequencies around u=0.26, so PhC periods are close to a=400nm. When the air filling factor is around 35%, the holes are around 200–250nm in diameter. Vertical stacks for active functions are larger than 3µm (each buffer layer for carrier injection has to be at least 1.5µm thick in order to limit the interaction of the mode optical field with the metal of the electrode). The aspect ratio of the holes then has to be larger than 10, and closer to 20 – a real challenge in InP-based materials. 2.3.1. Mask generation Nanometer scale features are easily fabricated using electron beam lithography. This method is very versatile, and enables the position of the holes to be changed on demand in order to investigate new structures. When structure designs are established, a lithographic mask can be generated, and deep UV lithography (248nm, 196nm) is then successfully used [BOG 02]. The advantage of such a process is that it implements the fully established and highly parallel microelectronic lithography. Having investigated advanced structures, we have generated here all the patterns using e-beam lithography. The mask used for etching the semiconductor material is realized in two steps, described in Figure 2.3a: first, a dielectric layer is deposited on the semiconductor material (here we use SiO2), and then an electron sensitive resist such as PMMA (PolyMethylMetAcrylate) is spin-coated on top. PMMA is a positive resist. Holes are exposed. After development (the developer is here a solvent for the monomer which is generated when the impinging electrons break the polymer links), patterns are holes in the PMMA layer. This mask is then used to dry-etch the SiO2 layer. When holes are correctly transferred to the underlying SiO2 layer, the remaining PMMA is removed by plasma-ashing. This two-step mask generation is necessary as the PMMA resist is very fragile and is difficult to use for semiconductor etching (work on this area is in progress).

Bidimensional Photonic Crystals

91

2.3.2. Dry-etching of InP-based semiconductor materials Generally speaking, semiconductor materials can be etched either by wet etching – electrophile agents are then used which reveal crystallographic planes, or by dryetching – which is a combination of physical and chemical etching within a plasma containing chemically active species and ions. Part of the etched products is evacuated by sublimation. Chlorine gas is commonly used for III–V materials plasma etching. Here we perform dry-etching since etching holes requires a strongly anisotropic process. The larger problem when etching holes versus other sub-micrometer features such as micro-pillars or line gratings is that it is difficult to evacuate etched products from the bottom of deep holes. Because sublimation of etched products plays an important role, InP-based materials are much harder to etch than GaAsbased materials: in the case of GaAs, the etched products have comparable volatilities (Ga and As are on the same line of the periodic classification), whereas in the case of InP, the etched products have wildly different sublimation temperatures: in the case of chlorine gas, PCl3 is volatile at room temperature, while a temperature of 200°C must be reached to evaporate InCl3 in the same pressure conditions. Temperature is in this case a critical parameter of the etching process. Due to the easier technology for patterning GaAs-based materials, all the fundamental PhC investigations in the 1990s were performed on GaAs structures. Plasmas which are made denser than the one obtained in a classical Reactive Ion Etching (RIE) – Capacitive Coupled Plasma (CCP) – greatly improve the etched depth in the case of the geometry of the holes. These dense plasmas are generated either by Inductive Coupled Plasma (ICP) or by Electron Cyclotron Resonance (ECR). Another deep-etching technique combines Ion Beam Etching (IBE) with a chemical reaction when injecting Cl2 gas directly at the surface of the sample: this is then called Chemically Assisted Ion Beam Etching (CAIBE). This technique is still a lab technique; results presented here were obtained in the laboratory of KTH, Kista (Sweden). For the removal of etched products, the temperature is raised to 230°C [MUL 02]. Figure 2.3b shows CAIBE etched holes deeper than 3µm. These holes are not perfectly cylindrical, and it is difficult to estimate the roughness of the etched surfaces. Optical characterization is needed to assess the air filling factor value correctly, and also the quality of the etched surfaces.

92

Nanophotonics

Figure 2.3. A: Mask generation in a dielectric layer using electron beam lithography; B: CAIBE etched holes in a GaInAsP semiconductor layer and buffer

2.4. Modal behavior and performance of structures 2.4.1. Passive structures It is very important to grasp the monomode/multimode behavior of a PhC waveguide in order to understand the spectral performance of more complex PhC structures. The characterization method used here is the so called end-fire method: the light generated by an external tunable monomode lazer is injected through a micro-lensed fiber in the device. Note that this device is transparent at these wavelengths. The polarization of the injected light is maintained under tuning. The spectrally resolved transmission is measured on a wavelength domain ranging from 1,410nm to 1,590nm. Fourier transforming the transmission spectrum enables us to identify all the optical cavities existing within a structure, which are revealed by the interferences of the coherent light. The fringe contrast of each cavity is related to the internal optical losses and to the reflection coefficient amplitude of the two

Bidimensional Photonic Crystals

93

reflectors that limit that cavity. We systematically use this fringe contrast measurement technique to characterize each cavity [TAL 03b]. 2.4.1.1. Straight guides, taper 2.4.1.1.1. Straight guides All Wi PhC waveguides, except W1, are multimode. In order to measure propagation losses, one must know on which mode propagation occurs. In order to undoubtedly inject the light on the fundamental mode of a Wi guide (and symmetrically collect the light exiting on the fundamental mode), each Wi is inserted between two deep ridge access guides which are monomode with a comparable modal profile. These ridges are realized during the same e-beam lithography step performed for the holes, thus their axes are aligned with the Wi guides’ axes. The Scanning Electron Microscope (SEM) picture in Figure 2.4a shows a top view of the ridge access guide on the left (the white area in between two dark lines) and a W2 ΓK waveguide on the right. The overall cavity formed by the input ridge waveguide + Wi + output ridge waveguide is limited by two cleaved facets. After calibration of propagation losses within the ridges and the reflection coefficient of the cleaved facets, measuring the transmission for different lengths of the Wi guide reveals the propagation losses within the Wi guide, and the reflection at the interfaces’ ridge/Wi [TAL 01]. Figure 2.4b displays the variation of propagation losses (overall losses including in-plane and out-of-plane losses) versus Wi guide width, reported here through the number i of missing rows. Large losses in the case of W1 can be attributed to the strong interaction of the field with the innermost row of holes. For larger Wi, losses are reduced, and we can expect that losses will continuously reduce when i is increasing. But, the loss figure obtained for W7 is of the same order as the one obtained for W5. We can demonstrate (through the width of the peak in the Fourier Transformed spectrum) that when propagating in a W7, even when injecting on the fundamental mode, higher order modes are excited. So part of the light is propagated on these modes, which are more lossy, leading to an overall propagation loss value larger than that which could be expected for a propagation performed only on the fundamental mode. Following the dispersion curve of W1 (Figure 2.2b), one can see that the even mode is slowed down when approaching the Brillouin zone edge. In a certain frequency range, no transmission is possible as no mode exists, and for even smaller frequencies, light propagation is again achieved through the continuum of optical modes. The spectral position of this cut-off is strongly dependent on the air filling factor ff. Figure 2.4c plots this cut-off for two different ff values. This optical method allows the determination of ff with an accuracy better than 1% [TAL 04b].

94

Nanophotonics

Turning to the multimode W3 waveguide, we refer again to the dispersion curve in Figure 2.2b. Figure 2.4d displays the transmission spectrum of a W3: the measured transmission is shown by a solid line, an FDTD simulation without losses is displayed as a dashed line, and a simulation including losses through the ε” parameter is plotted in bold dashed lines, here ε”=0.10 [SWI 01]. The mini stopband observed on the transmission spectrum is a signature of this waveguide. The ε” parameter is a phenomenological parameter whose value is adjusted to fit the measured data. The smaller ε”, the lower the losses. We recall here that above the substrate light-line, losses always exist through coupling vertically to radiative modes.

Figure 2.4. A: photonic crystal guide: Scanning Electron Micrograph (SEM) top view of a ridge access guide and a W2; B: Propagation losses versus PhC guide width, for Wi; C: Cut-off of W1 mode, versus ff (air filling factor); D: Mini stopband in a W3

2.4.1.1.2. Taper W1 waveguide is lossy, but has the advantage of being monomode. Monomode waveguides are the preferred choice for Integrated Optics devices as they prevent mode mixing; we make this choice too, and try to use W1, but on limited sections in order to minimize overall cumulative propagation losses. Such large losses for the W1 fundamental even mode stem from strong lateral confinement: Figure 2.5 displays the transverse modal profile of the fundamental mode of a ridge guide having the same confinement as the W1 mode in the refractive region. Due to this

Bidimensional Photonic Crystals

95

strong lateral confinement, the mode is laterally squeezed, and thus is greatly extended vertically in the case of the low index contrast InP-material system. Such a modal shape is not at all well adapted to that of a classical monomode ridge waveguide typically 1.2 to 1.5µm wide. A W3 waveguide is wider, and its fundamental mode is correctly adapted to the mode of a classical ridge access guide [TAL 01]. To reconcile both aspects, an original geometry has been proposed for a taper which adapts the fundamental mode of W3 to the fundamental mode of W1. This geometry is based on a continuous variation of the holes’ size and depth, thus adapting the mode in both directions of the transverse plane [LAL 02]. In theory, such a geometry enables almost 100% of the light from a W3 to a W1 to be coupled, provided that we are able to produce holes with a diameter as low as 70nm.

Figure 2.5. A: Optical mode confinement in a “W1-like” environment; B: SEM top view of the taper, and cross view of the first hole

Figure 2.5b displays the SEM picture of the first hole of the taper. In this fabrication run the diameter is 180nm and thus is still too large. Recent results have been obtained with a 100nm diameter hole within the taper; the coupling efficiency in that case is as high as 70% on a spectral domain larger than 80nm below the odd mode, and reflection is limited to 1% [TAL 04a].

96

Nanophotonics

2.4.1.2. Bend, combiner 2.4.1.2.1. Bend The main drawback of bends realized by index contrast is the large radius of curvature required for low bending losses, which leads to very extended devices. In the case of shorter bends (large curvature), polarization conversion also occurs [VAN 96]. PhC bends have the advantage of very short bends without in-plane losses. Being in the photonic gap, there is no possibility of coupling with in-plane radiation modes. The bend geometry has to be optimized to reduce and even cancel the reflection, which is the only in-plane mechanism that can reduce the transmission. Several geometries have been investigated when moving holes in the very corner. Partly optimized designs exist for each Wi width [OLI 02a]. But one should pay attention to two points: 1) After the bend, the light has to exit just as it entered, on the fundamental mode; this makes it possible for bends to cascade, or for a bend to be included in a more complex PIC. 2) The polarization must not have been altered. The bend breaks the symmetry of the crystal, so as soon as the guide is multimode, the bend projects the fundamental mode on all the modes supported by the guide [BEN 02, MEK 96]. FDTD simulation gives a clear view of a multimode versus a monomode propagation at the output of the bend. This multimode behavior is delicate to evidence from the far field pattern. To follow the triangular lattice geometry, bends turn at 60°, so a practical fabricated structure with cleaved parallel facets has to include two consecutive bends. We address here point 2, investigating the effect of the bend on the polarization state. Figure 2.6 shows SEM pictures of two cascaded bends for W3 and W1 waveguides, again with the transmission spectra. For each PhC guide width an optimized geometry has been implemented: it corresponds to six holes moved in the case of W3, and one hole moved in the case of W1. The external source uses a polarization maintaining fiber with a polarization rejection rate higher than 17dB. The light is injected on the TE polarization, using a polarization maintaining fiber, and is analyzed at the output of the collecting fiber through a polarizer. This enables the power level at the output to be measured for both polarizations. For W3 bends, we can see in Figure 2.5a that the power level on TM is only 10dB smaller than the one on TE: this means that part of the light injected on the TE polarization is now collected on TM; some polarization conversion has occurred in this bend. In the case of W1 bends, the measured TM level at the output is 17dB

Bidimensional Photonic Crystals

97

lower that the TE one, which is the source rejection. No polarization conversion (or below 17dB) has occurred. In conclusion we could say that W1 is lossy, but it is monomode and no polarization conversion occurs in the bend, whereas W3 has lower propagation losses, but some polarization conversion occurs in a bend. 6 holes moved

1 hole moved

Figure 2.6. Double bend in W1 and W3 PhC guides: on the left, optical transmitted power versus wavelength for both TE and TM polarizations; on the right, an SEM picture of each double bend

The best bend design ideally combines a wide PhC guide, e.g. W3 in the straight sections (for low propagation losses), and a narrow PhC guide, e.g. W1 in the very bend, which does not produce polarization conversion, and excites only the fundamental mode of the wide guide at the exit. We have realized such a bend including tapers between the straight W3 sections and the W1 bend [TAL 04a]. The FDTD simulation in Figure 2.7a shows that the light exits here on the fundamental mode of the W3. The fabricated structure (SEM picture in Figure 2.7b with the detail of the bend and the access taper) demonstrates losses as low as 1.5dB

98

Nanophotonics

per bend, when compared to a straight deep ridge (plotted as a dashed line), on a 30nm wide spectral range. It is clear from the SEM picture that there is room for improvement in the fabrication – taper holes have to be as small as 70nm in diameter, as previously stated. Performance will consequently be improved.

Figure 2.7. A: FDTD simulation (2D+ε”) of a bend including a taper; B: SEM pictures: general overview, and detail of the fabricated bend; C: Measured optical transmitted power – the straight guide is plotted as a reference

2.4.1.2.2. Y branching combiner Optical signal processing will require more complicated circuits such as interferometers (Mach-Zhender …). A generic building-block is a combiner, in which at least two optical paths have to be combined (divided). Figure 2.8a is an SEM picture of a two-branch combiner fabricated with W1 guides and including a taper at each input and at the output. Different geometries have been investigated for the combining region. The design rule followed here is to limit the excitation of higher order modes in the enlarged section. The best result is the one obtained on the structure shown on the enlarged SEM picture of the combining region, when adding a single hole of the same size with respect to a canonical design. Symmetry considerations are used to optimize this design.

Bidimensional Photonic Crystals

99

Transmissions plotted in Figure 2.8b show that both entrances C1 and C2 have the same power budget, which is 3dB lower than that of a bend [TAL 04b]. This is the same performance as a “classical” Y junction, but for a combiner which is 100 times smaller (104 in area footprint). The remaining oscillations on both the C1 and C2 spectra arise from the interference fringes due to the reflection at the cleaved facets which have no anti-reflective coating. The cut-off at 1,540nm is the signature of the W1 guide.

Figure 2.8. A: SEM pictures of a Y combiner: general overview, and detail of the combining section; B: Measured transmission spectra for both entrances – the bend alone is plotted as a reference

General rules for designing PhC guiding structures are based on symmetry considerations and modal shapes. It is not possible here to resolve the inverse problem as it is commonly done for 1D Bragg mirrors, where the stack of layers can be calculated to provide a spectral reflectance defined beforehand. Attempts are now made to obtain the best geometry using genetic algorithms: the parameters are the holes’ positions and sizes, a figure of merit is defined, and the algorithm produces the geometry that allows the best fit for this figure. Very little understanding of PhC physics is required to define such structures, though some optimal guiding geometries have been produced [HAK 05]. This approach is

100 Nanophotonics

probably time consuming, and is limited to guiding structures. Resonant structures, as we will see below, are based mainly on mode shaping. 2.4.1.3. Filters Let us recall that in Wavelength Division Multiplexing (WDM) networks, wavelengths are typically spaced at 0.8nm intervals (100GHz at 1.55µm), or even lower intervals (from 50 to 25GHz). A directive filter makes it possible to add or drop one wavelength selected among all the transmitted wavelengths. The filter rejection has to be better than 25dB. These requirements can be fulfilled for a filter having a quality factor Q larger than 2,000. The designs of filters, including PhC structures, can be based on the codirectional coupling mechanism, as in classical filters, channeling the light from one waveguide to another through a resonator. Specific designs implementing the contra-directional coupling that exists in PhC waveguides can also be produced. 2.4.1.3.1. Co-directional coupling Before fabricating a fully-fledged filter (guide + resonator + guide), we will investigate a drop structure which includes a cavity coupled to a guide. We want to demonstrate that one wavelength, which corresponds to a resonant mode of the cavity, is missing after transmission. Figure 2.9a shows the SEM picture of an H2 cavity coupled to a W3 guide. The transmission spectrum, after filtering the fringes due to the overall resonator limited by the two cleaved facets, shows a dip at 1,440nm, and the quality factor is 200. The light is co-directionally coupled from the refractive mode of W3 to the whispering gallery mode of H2 (see section 2.2.2, Figure 2.2a). The filter in Figure 2.9b couples an H7 cavity to a W3 guide. The epitaxial layers used for this design include quantum boxes on GaAs whose luminescence is excited by optical pumping. In the case where the cavity is separated from the guide by two rows of holes, the light is extracted from the guide, with a quality factor of 800 (spectrometer limited), through the slow mode of the mini stopband, as represented by the dark arrows [SMI 00].

Bidimensional Photonic Crystals

101

Figure 2.9. A: Guide-cavity coupling through the gallery mode; B: Guide-cavity coupling through the slow mode of the mini stopband

2.4.1.3.2. Contra-directional coupling Within a multimode PhC waveguide, the mini stopband is the result of contradirectional coupling between two modes of this guide, through the periodicity of the PhC guide itself. When the design makes it possible to collect the light on the backward mode, this wavelength is dropped from the forward mode, over the spectral domain of the mini stopband. The structure proposed by Qiu et al. [QIU 03] includes 2 PhC guides: one is a W1, the other is a W0.8; both are separated by one row of holes (Figure 2.10a). Coupling between these two guides creates a mini stopband, visible on the dispersion curve. The optical spectrum measured on the collection guide (drop) is compared to the calculated one. Light is correctly transferred and the quality factor is Q=150, again limited by out-of-plane losses (device on InP substrate). Benisty et al. have proposed a structure based on the slow mode of the mini stopband of a W5 guide or similar (Figure 2.10b). This structure also includes quantum boxes on GaAs for characterization. Both transmissions measured on the incident guide (bar) and on the transferred guide (cross) are displayed as bold lines in Figure 2.10b; the calculated transmissions are plotted as grey lines [OLI 03]. The quality factor Q is 150, again limited by the out-of-plane losses.

102 Nanophotonics

All these results show that it is difficult to reach the desired Q value when operating on a substrate. FDTD simulations including out-of-plane losses through the ε” parameter clearly evidence that any efficient in-plane wavelength selective mechanism loses its effectiveness when including out-of-plane losses, even a low level of losses. On the other hand, quality factors as large as 100,000 have recently been reached for membrane structures on SOI material [AKA 05]. Design rules for efficient resonators are based on modal shape analysis taking into account the light cone.

Figure 2.10. A: Contra-directional coupling, through the mini stopband, in the W1/W0.8 system; B: Measured transmission of coupling through the mini stopband of W5

2.4.2. Active structures: lazers PICs fulfilling all the optical functions will require active functions, such as emission, amplification or fast tunability obtained through carrier injection or biasing. Injecting carriers or biasing a deeply etched structure with holes is very difficult.

Bidimensional Photonic Crystals

103

A lazer is a good example of an “active function”. The threshold current offers reliable information about surface recombination due to all the defects produced during etching, while external efficiency is related to optical losses for the lasing mode. One can also take advantage of the periodicity inherent in the PhC to reach monomode operation. Here we present results on full-PhC edge-emitting lazers, operating on continuous wave under electrical injection at room temperature. In the structure presented in Figure 2.11a, spectral selectivity is provided by coupling 40 cavities of size H7 [HAP 01], following the proposal of the so called CROW (Coupled Resonator Optical Waveguide) [OLI 01 and references therein]. Coupling such a large number of cavities ensures a very large side mode suppression ratio (SMSR) for the resonant mode. The design proposed in Figure 2.11b has no holes in the axis of the cavity. In this case, monomode behavior is achieved by adding another periodicity on the PhC matrix: one hole is added every six holes along the W5 ΓK guide. This additional periodicity folds the dispersion curve and allows the fundamental mode to be folded within the gap, to the 10th order. This behavior is “DFB-like”, but without losses on the other orders as they lie outside of the gap. This lazer also has a PhC rear mirror calculated to have its maximum efficiency at the lasing wavelength [LAB 99]. The measured SMSR was found to be 25dB, and the external efficient reached 0.17 W/A; these performances are comparable to devices actually implemented in optical networks [TAL 03a]. These results demonstrate that PhC structures are efficient for active optical functions. As amplification can be obtained on the same electrical scheme, a PhC amplifier can be included along with a passive PhC function, leading to a full PhCbased PIC demonstrating an overall power budget of 0dB.

Intensity, dBm

104 Nanophotonics

wavelength nm

Figure 2.11. Lasing operation of full photonic crystal lazers. A: Coupled cavity structure (CROW) showing a very large rejection; B: “DFB-like” lazer, showing a large rejection, and a large external efficiency η=0.17

2.5. Conclusion This chapter has discussed several passive and active devices based on photonic crystals. These devices can be introduced as building blocks in Photonic Integrated Circuits. Substrate-based structures are found to be adequate to obtain lasing emission or amplification, but when resonant performances are required, the limitation which is inherent as a result of losses through out-of-plane radiation modes is not acceptable for WDM applications. Photonic crystals also have interesting properties outside the photonic gap, related to the particular curvature of dispersion bands, such as the super-prism effect [KOS 98] or autofocussing. The nanotechnology required for these devices has been intensively developed, and has enabled the fabrication of structures including holes with diameters smaller than 100nm. Investigations are currently in progress to estimate the ultimate

Bidimensional Photonic Crystals

105

performances of PhC structures within the limits of today’s technology (lithography accuracy, etching quality). 2.6. References [AGI 03] Agio M “Optical properties and wave propagation in semiconductor-based twodimensional photonic crystals” PhD, University of Pavia, Italy [AKA 03] Akahane Y, Mochizuki M, Asano T, Tanaka Y and Noda S (2003) “Design of a channel drop filter by using a donor-type cavity with high-quality factor in a twodimensional photonic crystal slab” Appl. Phys. Lett. 82: 1341 [AKA 05] Akahane Y, Asano T, Takano H, Song BS, Takana Y and Noda S (2005) “Two-dimensional photonic-crystal-slab channel-drop filter with flat-top response” Optics Express 13: 2512 [BEN 96] Benisty H (1996) “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries” J. Appl. Phys. 79: 7483 [BEN 99] Benisty H, Weisbuch C, Labilloy D, Rattier M, Smith CJM, Krauss TF, De La Rue R, Houdré R, Oesterle U, Jouanin C and Cassagne D (1999) “Optical and confinement properties of two-dimensional photonic crystals”, J. Light Technol. 17: 2063 [BEN 00] Benisty H, Labilloy D, Weisbuch C, Smith CJM, Krauss TF, Cassagne D, Béraud A and Jouanin C (2000) “Radiation losses of waveguide-based two dimensional photonic crystals: positive role of the substrate” Appl. Phys. Lett. 76: 532 [BOG 02] Bogaerts W, Wiaux V, Taillaert D, Beckx S, Luyssaert B, Bienstman P and Baets R (2002) “Fabrication of photonic crystals in Silicon-on-Insulator using 248nm deep UV lithography” IEEE J. Selec. Top. Quant. Elec. 8: 928 [HAK 05] Hakansson A and Sanchez-Dehesa J (2005) “Inverse designed photonic crystal de-multiplex waveguide coupler” Optics Express 13: 5440 [HAP 01] Happ TD, Markard A, Kamp M and Forchel A (2001) “Single-mode operation of coupled-cavity lasers based on two-dimensional photonic crystals” Appl. Phys. Lett. 79:4091 [JOA 95] Joannopoulos JD, Meade RD and Winn JN (1995) Photonic Crystals. Princeton University Press, Princeton, NJ [KOS 98] Kosak H, Kawashima T, Tomita A, Notomi M, Tamamura T, Sato T and Kawakami S (1998) “Superprism phenomena in photonic crystals” Phys. Rev. B 58: r10097 [LAB 99] Labilloy D, Benisty H, Weisbuch C, Krauss TF, Cassagne D, Jouanin C, Houdré R, Oesterle U and Bardinal V (1999) “Diffraction efficiency and guided light control by two-dimensional photonic-bandgap lattices” IEEE J.Quant. Electron. 35: 1045

106 Nanophotonics [LAL 02] Lalanne P and Talneau A (2002) “Modal conversion with artificial materials for photonic-crystal waveguides” Optics Express 10: 354 [LOU 05] Lourtioz JM, Benisty H, Berger V, Gérard JM, Maystre D and Tchelnokov A (2005) Photonic Crystals: Towards Nanoscale Photonic Devices, Ed. Springer, Berlin, Heidelberg, New York [MEK 96] Mekis A, Chen JC, Kurland I, Fan S, Villeneuve PR and Joannopoulos JD (1996) “High transmission through sharp bends in photonic crsytal waveguides” Phys. Rev. Lett. 77: 3787–90 [MUL 02] Mulot M, Anand S, Carlström CF, Swillo M and Talneau A (2002) “Dry etching of photonic crystals in InP-based materials” Physica Scripta, T101: 106–9 [OLI 01] Olivier S, Rattier M, Benisty H, Weisbuch C, Smith CJM, De La Rue RM, Krauss TF, Oesterle U and Houdré R (2001) “Mini-stopbands of a one-dimensional system: the channel waveguide in a two-dimensional photonic crystal” Phys. Rev. B Brief Report 63: 113311 [OLI 02a] Olivier S, Benisty H, Weisbuch C, Smith CJM, Krauss TF, Houdré R and Oesterle U (2002) “Improved 60° bend transmission of submicron-width waveguides defined in two-dimansional photonic crystals” IEEE, J. Lightwave. Technol. 20: 1198–1203 [OLI 02b] Olivier S, Benisty H, Smith CJM, Rattier M, Weisbuch C and Krauss TF (2002) “Transmission properties of two-dimensional photonic crystal channel waveguides” Optical and Quantum Electronics 34: 171-181 [OLI 03] Olivier S, Benisty H, Weisbuch C, Smith CJM, Krauss TF, Houdré R and Oesterle U (2003) “All-Photonic crystal add-drop filter exploiting low group-velocity modes” WeB1.2, conference ECIO [QIU 02a] Qiu M (2002) “Effective index method for heterostructure-slab-waveguidebased two-dimensional photonic crystals” Appl. Phys. Lett. 81: 1163 [QIU 02b] Qiu M, Jaskorzynska B, Swillo M and Benisty H (2002) “Time-domain 2D modelling of slab-waveguide-based Photonic Crystal devices in the presence of radiation losses” Microwaves and Optical Tech. Lett. 34: 387 [QIU 03] Qiu M, Mulot M, Swillo M, Anand S, Jaskorzynska B, Karlsson A, Kamp M and Forchel A (2003) “Photonic crystal optical filter based on a contra-directional wavguide coupling” Appl. Phys. Lett. 83: 5121 [SMI 00] Smith CJM, De La Rue RM, Rattier M, Olivier S, Weisbuch C, Krauss TF, Houdré R and Oesterle U (2000) “Coupled guide and cavity in a two-dimensional photonic crystal” Appl. Phys. Lett. 78: 1487 [SWI 01] Swillo M, Qiu M, Mulot M, Jaskorzynsla B, Anand S and Talneau A (2001) “Characterisation and modeling of InP/GaInAsP photonic-crystal waveguides” ECOC 2001 [TAL 01] Talneau A, Legouezigou L and Bouadma N (2001) “Quantitative measurement of low propagation losses at 1.55µm on planar photonic crystal waveguides” Opt. Lett. 26: 1259–61

Bidimensional Photonic Crystals

107

[TAL 03a] Talneau A, Mulot M and Anand S (2003) “CW monomode operation of efficient full-photonic crystal lasers at 1.55µm” PDTh 4.2.1: 44–5, ECOC 2003; also Appl. Phys. Lett. (2004) 85: 1913 [TAL 03b] Talneau A, Mulot M, Anand S and Lalanne P (2003) “Compound cavity measurement of transmission and reflection of a tapered single-line photonic-crystal waveguide” Appl. Phys. Lett. 82: 2577–2579 [TAL 04a] Talneau A, Agio M, Soukoulis CM, Mulot M, Anand S and Lalanne P (2004) “High bandwidth transmission of an efficient photonic crystal mode converter” accepted at the PECS V conference, Kyoto, March [TAL 04b] Talneau A, Mulot M, Anand S, Olivier S, Agio M, Kafesaki M and Soukoulis CM (2004) “Modal behavior of single-line photonic crystal guiding structures on InP substrate” Photonics and Nanostructures – Fundamentals and Applications [VAN 96] Van Dam C, Spiekman LH, Van Ham FPGM, Groen FH, Van der Tol JJGM, Moerman I, Pascher WW, Hamamcher M, Heidrich H, Weinert CM and Smit MK (1996) “Novel compact polarization converters based on ultra short bends” IEEE Phot. Technol. Lett. 8: 1346–48

Chapter 3

Photonic Crystal Fibers

3.1. Introduction Thanks to the great strides taken at the very beginning of the twenty-first century, a large number of active and passive functions (emission, reception, filtering, wavelength multiplexing-demultiplexing …) can be operated by means of photonic integrated circuits (PICs). Thus, as demonstrated in the preceding chapter, the necessary basic elements are now available for designing advanced integrated functions for high bit rate optical transmissions. However, to take full advantage of this progress, high quality connections with optical fibers must be achieved. The transmission fibers themselves must also exhibit higher and higher levels of performance. The connection between PICs and standard optical fibers usually involves a taper for adjusting the size of the mode. This is followed by a diffraction grating engraved in the circuit for extracting light. The measured coupling efficiency is higher than 25% and can potentially exceed 80% [MCN 03; TAI 03]. A considerable increase in the transmission bit rate into one single optical fiber has already been made possible, thanks to two major breakthroughs over the past 20 years: first, since 1985, the use of fibers operating in the single mode regime in the spectral bandwidth of the highest transparency of silica (around 1,550nm), and secondly, since 1990, the all-optical amplification of spectrally multiplexed high bit rate signals into rare earth doped fibers. Nevertheless, there is still much to be done – in particular, increasing further the bit rate of optical communication around 1,550nm, and for many other applications Chapter written by Dominique PAGNOUX.

110

Nanophotonics

at different wavelengths, novel fibers must be designed in order to reduce, cancel or compensate for chromatic dispersion at the operating wavelength. Indeed, the only means of significantly changing chromatic dispersion is to act on the contribution of the guide to this parameter: this means that one must imagine fibers in which the confinement of the guided mode spectrally evolves in a radically different way from that observed in usual fibers. In order to meet this need, a new generation of optical fibers with specific micro or nano-structuration of the cladding, known under the generic designation of “photonic crystal fibers” (PCFs), has been conceived. As we will see below, a particular class of PCFs guides light by means of a photonic bandgap effect similar to that used in photonic crystals described in the preceding chapter. With these fibers, low loss propagation of light can be achieved in a hollow core provided that the submicrometric structuration of the cladding is perfectly controlled over long lengths (tens or hundreds of meters, indeed kilometers). These fibers are not designed to perform particular new functions, unlike PICs. Their main job is to provide propagation properties over long distances that cannot be obtained with other guides. In other words, they have a complementary role to play in the field of photonics. The history of photonic crystal fibers concretely begins in 1995, when a group of British researchers under the direction of Philip St J. Russell (Optical Research Center in Southampton, later University of Bath) decided to exploit, in the domain of optical fibers, the concept of photonic band gap already implemented for confining light in periodic bidimensional semi-conductor structures [BIR 95]. The sought result is a guiding effect due to a transverse Bragg resonance into the cladding of the fiber. In other words, it is a 2D application of the principle of Bragg fibers with 1D periodical cladding theoretically demonstrated by Yeh et al. in 1978 [YEH 78]. The optical cladding of Bragg fibers is made of a stack of transparent layers with alternately high and low indices. It can efficiently confine light in the core, even if the index of this core is lower than the minimum index of the layers of the cladding, since the usual total internal reflection principle is not involved in the guiding mechanism [BRE 00b; FIN 99]. In particular, the core may be filled with air (hollow core). Obviously, the transmission bandwidth of such a fiber is limited to the transverse resonance bandwidth of the cladding [BRE 00a]. For a given pair of materials constituting the layers of the cladding, the central wavelength of the transmission bandwidth decreases with the thickness of the layers. For example, light can be guided in the near infrared (around 1,550nm) in a hollow core Bragg fiber whose cladding is made of 35 alternate layers of glass (As2Se3) and polymer (PEI) having refractive indices respectively equal to 2.82 and 1.66 at 1,550nm. In

Photonic Crystal Fibers

111

order to guide light in this spectral bandwidth, the thickness of the layers must be equal to 270nm and 470nm respectively [KUR 04]. In Russell’s project, the concentric layers with alternate high and low indices constituting the resonant optical cladding of the Bragg fibers are replaced by a hexagonal or a triangular lattice of air holes running along the axis, in a pure silica medium (Figure 3.1). Because of the regular micro-structuration of the cladding, an optical wave launched into the core experiences a periodic modulation of the surrounding medium whatever the considered direction in the transverse plane. By analogy with photonic crystals designed to guide light into semi-conductor media, these microstructured fibers have been called “photonic crystal fibers”. However, this novel type of optical waveguide differs radically, due to the fact that the light does not propagate any more perpendicularly to the invariant direction of the lattice (z direction), but parallel with this direction. This peculiarity must be pointed out because photonic bandgaps in fibers made of silica and air could not exist in other conditions. Indeed, in the case of a wave propagating perpendicularly to the hole axis, one could obtain a photonic bandgap whatever the polarization only if the air fraction (denoted f) was higher than 0.66 and, above all, if the index contrast was at least equal to 2.66; that is to say if it was higher than the 1.45 index contrast existing between air and silica [VIL 92]. These requirements are considerably reduced if we consider an optical wave whose wave vector has a non-zero axial component β. In this case, with an index contrast equal to only 1.45 and a sufficient air fraction, there are bandwidths where, for a given value β, one cannot find any pair of transverse components (kx , ky) whatever the considered polarization. The propagation of an optical wave at a wavelength in one of these bandwidths, through the periodic structure, is thus impossible: these bandwidths are photonic bandgaps. Then, if a defect set into the structure (hole with a size different from the others, or lake of hole) allows the existence of a mode with a propagation constant β, this mode must remain confined to this defect that behaves as the optical “core” of the fiber (Figure 3.1).

112

Nanophotonics

silica air

Triangular lattice

Hexagonal lattice

Photonic crystal fibres

Figure 3.1. Cross-sections of photonic crystal fibers based on a triangular or hexagonal lattice of holes

3.2. Two guiding principles in microstructured fibers The two main geometrical parameters of photonic crystal fibers are the diameter of the holes (d) and the spacing between two adjacent holes, called the “pitch” and denoted by Λ. These two dimensions are related to the air fraction f by f≈0.91(d/Λ)2. The photonic bandgaps in the cladding of a triangular lattice air/silica photonic crystal with a 45% air-filling fraction can be identified in Figure 3.2 by means of kΛ(grey zones), where k is the modulus of the wave vector in the vacuum [BIR 95]. These wavelengths are those likely to be guided into the core of the fiber. In the considered example, they exist for values of β greater than 7/Λ. However, it is necessary to distinguish two types of spectral bandwidths whose wavelengths cannot propagate into the cladding. The first one (the grey triangle on the right at the bottom of Figure 3.2) corresponds to wavelengths guided into the core by virtue of the total internal reflection principle. These wavelengths comply with the relation β>k0neg where neg is the effective index of the fundamental mode able to exist in the heterogeneous cladding whose extension is supposed to be infinite. neg, which for simplicity’s sake will be referred to as “cladding effective index”, is given by:

2 n eg =

∫∫

∫∫ n E dS − 2 2 2 ∫∫ E dS k ∫∫ E dS 2

2

2

dE dS dr

(1)

Photonic Crystal Fibers

113

where E is the electric field; n is the index of the silica (ns) or of the medium filling the holes (na) depending on the considered point of the cross-section; S is the area of an elementary cell of this cross-section; and r is the distance to the center of the fiber. For these wavelengths there exists no possible refraction angle between the core and the cladding. In other respects, there are also four other narrow bandwidths, at shorter wavelengths, due to the true guiding by the Bragg resonance in the photonic crystal. These bandwidths are obtained for all the polarizations of the electromagnetic field, for values of the product kΛ comprised between 8 and 12. Thus, for obtaining bandwidths in the near infrared or the visible, the pitch must be 1 to 2 µm. This means that it is far larger than that necessary in planar guided optics. The explanation lies once again in the fact that light propagates in the direction of invariance of the crystal and not following the plan of periodicity [BIR 04].

Normalized frequency kΛ

12 11 10 9 8 7 6 6

7 8 9 10 11 Normalized propagation constant βΛ

12

Figure 3.2. Photonic bandgaps in a triangular lattice air/silica photonic crystal with a 45% air-filling fraction [BIR 95]

The very first fiber manufactured to obtain a guiding effect in an air/silica photonic crystal consisted of one silica rod surrounded by narrow air holes with somewhat different diameters (0.2 to 1µm), spaced at approximately 2.3µm intervals. A single mode was guided at different wavelengths from 337nm to 1,550nm, so one could think that the attempt was successful [KNI 96]. However, more careful study shows that this fiber cannot exhibit photonic bandgaps because the air-filling fraction into the cladding is too weak. The single mode guiding observed over a surprisingly large bandwidth is in fact easily explained by the two following considerations: – neg, the effective index of the heterogeneous medium constituting the optical cladding, that is the result of a balance between the index of the air and the index of the silica, is necessary lower than that of the pure silica core (see section 3.4). The observed guiding effect is then simply due to the total internal reflection (TIR) at the

114

Nanophotonics

frontier between the cladding area and the core area whose limits are not physically located. The fiber behaves as a step index fiber with a numerical aperture equal to NA=(ns2-neg2)1/2. The core radius can be stated as aeq=0,64Λ: indeed, one can show that the propagation constants of the modes of the step index fiber described by these opto-geometrical parameters are very close to those of the corresponding modes of the microstructured fiber [BRE 00c]; – as the extension of the field into the holes depends heavily on the wavelength, one can suspect that, considering equation 1, neg do not vary conventionally with respect to the wavelength. For example, for wavelengths significantly longer than d and Λ, the field largely spreads into the holes and, in a first approximation, it can be considered as a plane wave propagating into a medium having its index equal to neg=[f.na2 + (1-f).ns2]1/2. On the contrary, the shorter the wavelengths, the more the field avoids the air holes so that the effective index of the cladding asymptotically tends towards that of silica, under the form [REE 02]:

n eg

2 ⎛ d ⎛λ⎞ ⎞ ≈ ⎜ n s2 − ⎜ ⎟ ⎟ ⎜ Λ ⎝ Λ ⎠ ⎟⎠ ⎝

1/ 2

(2)

The above relation shows that the numerical aperture proportionally decreases with the wavelength. The spatial normalized frequency V=k.aeq.NA, that is in fact the Fresnel number in one dimension associated to the guided beam divided by 2π, gives one an indication of the number of spatial samples in this beam, that is to say, of the number of modes that can be guided. V tends from lower values towards a limit Vlim that depends only on the ratio d/Λ (Figure 3.3). When Vlim is lower than the cut-off spatial frequency of the second mode (2.405 when aeq=0.64Λ), the fiber remains theoretically single mode whatever the wavelength [BIR 97]. This property can be fulfilled even with very large core fibers [KNI 98a]. However, as the effective index of the cladding approaches that of silica, the efficiency of the guiding decreases and the bend loss increases [SOR 01]. This drawback limits the single mode bandwidth that can actually be used. The non-usual spectral dependence of the effective index of the cladding, due to the heterogeneous nature of the medium, also strongly influences the effective index and the chromatic dispersion of the fundamental mode propagating into the core. By choosing d and Λ carefully, one can adjust the effective area of the mode over a large range of values. The birefringence may also be adjusted by acting on the symmetry of the structure. Considering these features, one can easily understand that microstructured fibers guiding light due to the total internal reflection principle exhibit attractive propagation properties, even if the well known guiding principle has been exploited for years in standard fibers. As we will see below, these

Photonic Crystal Fibers

115

properties open the way for novel applications in very diverse domains such as telecommunications, metrology, spectroscopy and non-linear optics. This is the main reason for the intensive research that has been undertaken since 1996 in numerous laboratories in the industry (Corning, Lucent Technologies, Alcatel, NTT-Ibaraki …) and in the universities (University of Bath, ORC in Southampton (UK), TUD Lyngby (Denmark), IRCOM Limoges (France), University of Sydney (Australia) …). Bodies such as Crystal Fiber (Denmark), specially devoted to their fabrication and distribution, have also been created in recent years. It is natural to devote space to them in this chapter.

Multimode region

Normalized spatial frequency V

2.5

d/Λ=0.4

2,405

2

d/Λ=0.3 d/Λ=0.2

1.5

d/Λ=0.1 1 0.5

aeq=0.64Λ

0 0

1

2

3

4

5 6 aeq /λ

7

8

9

10

Figure 3.3. Spectral dependence of the spatial normalized frequency showing that the microstructured fibers remain single mode whatever the wavelength for weak values of the d/Λ ratio

Before continuing, it is worth noting that the label “photonic crystal fiber”, wrongly attributed to the very first microstructured fiber from the University of Bath in 1996, has since been used in most publications to refer to the microstructured fibers that guide light following the total internal reflection principle. In this chapter, to avoid confusion, such fibers will be called “air/silica microstructured optical fibers” or, for simplicity’s sake, “TIR-MOFs”. The label “photonic crystal fibers” (PCFs) will be reserved for those guiding due to a photonic bandgap (PBG) in the cladding.

116

Nanophotonics

3.3. Manufacture of microstructured fibers The technical process generally implemented in manufacturing microstructured fibers (TIR-MOFs or PCFs) is quite similar to that used for conventional silica fibers. In both cases, the fiber is drawn at a high temperature (between 1,800 and 2,000°C) from a preform set in a vertical furnace at the top of a drawing tower. However, preforms are elaborated following very different techniques. On the one hand, the preform of a usual fiber can be described as a pure silica rod with a diameter of few centimeters, in the center of which a doped silica region is incorporated. Due to the dopants (Al, Ge, P …) the index of the doped region is higher than that of the silica, and this region is intended to become the core of the fiber. The creation of this kind of preform necessitates the implementation of complex physico-chemical techniques, such as MCVD, OVD or VAD. The index profile of the drawn fiber is a homothetical reduction of that of the preform. Preforms of MOFs are obtained by carefully stacking an assembly of pure silica capillary tubes and rods having a diameter of about 2mm. This assembly is inserted into a maintaining tube that constitutes the initial preform whose diameter is a few centimeters (Figure 3.4a). The general arrangement of the lattice of holes in the preform, i.e. the number and distribution of these holes, must be maintained throughout the drawing process. But contrary to the case of classical fibers, the final cross-section of an MOF does not systematically result from a simple homothetical reduction of that of the preform. Numerous parameters, including the temperature of the furnace, the pressure inside the tubes and the speed of the drawing operation, have a significant influence on the shape and size of the holes in the fiber, and on those of the possible residual interstices between these holes (Figure 3.4b and c). The used methods, so called “stack and draw methods”, enable a large variety of MOFs to be manufactured; they are characterized by the distribution of the holes in the lattice (triangular or hexagonal), the dimension of the core, the air-filling fraction (in other words the ratio d/Λ) and the number of rings of holes. These parameters are chosen with respect to the targeted application. In order to obtain performing MOFs, the following conditions must be met: – very high purity silica rods and tubes, perfectly desiccated, must be used, with the lowest possible rugosity at the silica/air interface, to reduce propagation loss; – the nominal dimensions of d and Λ must be respected because the effective area of the guided mode, its chromatic dispersion and the birefringence are very sensitive to these parameters; – the cross-section must not be varied axially – this is absolutely vital for achieving low propagation losses, especially in the case of hollow core PCFs [VEN 02].

Photonic Crystal Fibers

a)

b)

117

c)

Figure 3.4. Photographs of cross-sections: A: a preform of an MOF; B and C: an MOF drawn with unsuitable (b) and inadequate(c) drawing parameters (photographs from IRCOM)

3.4. Modeling TIR-MOFs As is the case whatever the type of guide, the main goal when modeling the propagation of light into microstructured fibers is to determine the effective index of each mode together with the spatial distribution of the associated electromagnetic field. The chromatic dispersion is deduced from the variations in the effective index versus the wavelength, whereas the field distribution enables the effective area, the confinement loss and the splice loss to be calculated.

3.4.1. The “effective-V model” With the so called “effective-V model”, one seeks at each wavelength the optogeometrical parameters of a step-index fiber whose propagation characteristics are identical to those of the studied MOF (equivalent step-index fiber) [BIR 97]. The index of the core of this fiber is that of pure silica, ns. The index of the cladding is the highest effective index associated with a mode able to propagate in the silica/air structure when the latter is supposed to be infinite and free from all geometrical defects. This mode is known as the “fundamental space filling mode”. Among the modes of the infinite structure, the fundamental space filling mode is the one with the largest part of its energy located in the silica. One calculates neg by solving the Maxwell equations over an elementary cell that is a rectangle whose two opposite angles are the centers of neighboring holes. By duplicating this cell using symmetries versus its edges, the whole silica/air structure is reconstituted. The effective indices of the modes of the MOF are very close to those of the corresponding modes of the equivalent step-index fiber having a core radius equal to 0.64Λ. Thus, by replacing the MOF with its equivalent step-index fiber at each

118

Nanophotonics

wavelength, one can easily determine the spectral single mode propagation domain, the chromatic dispersion and the bend loss of this MOF. The effective-V model is very easy to implement but, as it cannot bring about distribution of the guided mode, it is of limited utility. 3.4.2. Modal methods for calculating the fields Modal methods consist of calculating the components of the electromagnetic fields of the guided modes due to the resolution of the wave equation, taking into account the actual index profile of the fiber that is supposed to be invariant along its axis. Generally, vector formulations are preferred to scalar ones, which are acceptable only if the index difference between the core and the cladding is very low (weak guidance approximation). This condition is rarely respected in MOFs, especially those with a large air-filling fraction in the cladding. One method involves solving the wave equation by replacing the expressions of the electromagnetic field and the index profile by their decomposition over a properly chosen base of functions. They can be: – trigonometric functions (Galerkin method) [KIM 00]; – cosine functions for the periodic region of the index profile and Hermite-Gauss type localized functions for the central area and the fields [MON 99, 00]; – Hermite-Gauss functions exclusively for both the index profile and the fields. This last choice is generally the most judicious because the decrease in the field is suitably described by means of a minimal number of localized functions, allowing a quick convergence of calculations and a lower memory requirement of the computer. The finite element method is based on the resolution of the Maxwell equations at each node of a triangular grid previously operated over the index profile of the fiber [NED 86]. In order to guarantee a proper description of the fields, the regions close to the core where the amplitude variations are significant necessitate fine mesh (length of the edge of a mesh between two successive nodes 0.81) [POT 03]. As expected, the confinement loss of the mode is reduced as the air-filling fraction in the cladding is increased and as the number of rings of holes is increased. To indicate some orders of magnitude, it is divided by 1,000 when f increases from 73% to 80%; for f = 80%, it decreases from 100dB/m to 0.01dB/m when the number of rings of holes is increased from 4 to 12 [SAI 03]. At least, the shape of the central hole has a great influence on the confinement loss. It is shown that this loss is significantly reduced if the cross-section of the central hole is close to that depicted in Figure 3.6, instead of a perfect hexagon.

126

Nanophotonics

a)

b)

Figure 3.6. Cross-section of a hollow core PCF. A: general shape [SAI 03]; B: fiber manufactured at the University of Bath [CRE 99]

The very first demonstration of guiding in a hollow core PCF, in 1998, was not really convincing because the holes were organized following a hexagonal matrix that allowed the field to spread out of the core and relocate in the silica interstices at the close periphery of this core [KNI 98b]. On the contrary, a triangular lattice of holes effectively confines the field in the hollow core [CRE 99]. The very high loss measured in the first manufactured PCFs (1999–2000) were essentially due to a lack in the axial uniformity of the fiber that randomly shifts the position of the bandgap along the propagation [WES 00, 01]. A second cause of loss is the coupling of light from the core to the silica (surface modes) that is made minimal by the use of a large section hollow core surrounded by a microstructured cladding with very narrow silica partitions [DIG 04]. The thickness of these partitions must remain smaller than λ/10. Over the course of a few years, advances in manufacturing techniques have made it possible to reduce propagation loss considerably [VEN 02; HAN 03; MAN 04]. Figure 3.7 shows the cross-section of a PCF whose propagation loss is reduced to less than 2dB/km over a 10nm spectral width around 1,565nm, due to scrupulous adherence to the above conditions. The elementary motives in the cladding have a 3.9µm diameter and the thickness of the silica partitions is about 120nm. These promising results allow one to dream that, in the near future, the loss of hollow core PCFs will be reduced, over a large spectral range, under the 0.15 dB/km threshold imposed by Rayleigh diffusion in conventional fibers. In such fibers, the power threshold over which significant non-linear effects appear should be 100 times higher than in usual fibers. They are obviously of great interest for the transmission of signals at wavelengths strongly absorbed by the silica (UV and far IR) and for the guiding of high power pulses without spectral distortions.

127

Attenuation (dB/km)

Photonic Crystal Fibers

a)

b) Wavelength

Figure 3.7. A: Microscope Electronic Beam picture of the cross-section of a PCF whose core is realized by removing the central rod and the two first rings of capillaries in the structure; B: spectral attenuation of this fiber [MAN 04]

3.6.2. Some applications of photonic crystal fibers It is well known that very small objects (from atoms to micro-particles) can be trapped in the area of highest power density of a light beam and moved under the action of the radiation pressure of light. By launching the trapping beam into the hollow core of a PCF, micro-objects can be guided over a long distance in the center of this core, while following the curvature of the fiber, without any contact with the lining, and directed precisely towards the desired destination. In the first experiment that has proven the validity of this process, 5µm diameter polystyrene balls were guided by a continuous beam from an argon lazer (λ=514nm, P=80mW) over 15cm at a speed of 1cm/s in the center of the 20µm diameter core of a PCF (d=3–4µm, f=75%, attenuation=5dB/m at 514nm) [BEN 02b]. The length over which particles are guided can potentially exceed 100 meters in a fiber with lower attenuation, such as that shown in Figure 3.7. This opens up new horizons for all the applications that necessitate moving micro-objects to a precise location without any contact: for example, biology (handling of viruses and bacteria), chemistry and atomic physics. By filling the hollow core of a PCF with a highly non-linear gas, one can obtain significant effects with a very low pumping level, due to the very long lengths of interaction between the light and the gas along the fiber. As an example, in the core of a PCF filled with hydrogen under pressure, a pulsed lazer beam generates Stokes and anti-Stokes lines with a necessary energy in each pulse two orders of magnitude lower than that required by other techniques (interaction in a capillary or in the waist of a focused beam). With pulses of only 800 nJ/6 ns, a 30% conversion efficiency in

128

Nanophotonics

the Stokes line is measured at the output of only a few centimeters-long sample of PCF [BEN 02a]. A PCF can also guide, over several meters, femtosecond soliton pulses with a peak power of several megawatts. In order to achieve this, certain conditions must be fulfilled. First, the chromatic dispersion of the mode of the fiber must cancel at a wavelength close to the central wavelength of the pulse. Second, the propagation must operate in a material having a very low non-linear coefficient. Finally, no Raman effect may appear because it should quickly shift the spectrum of the pulse out of the transmission bandwidth of the fiber. These conditions are met in a PCF with a 13µm diameter core (f=94%) filled with rare gas (xenon) and with its chromatic dispersion cancelled at 1,425nm [OUZ 03]. Gaussian soliton pulses (5.5 MW/75 fs), centered at 1,470nm, have been demonstrated to be properly guided over a 2m length of fiber. The applications are principally in the domains of spectroscopy, biology and medicine. Finally, in order to achieve novel functions by means of fibers similar to PCFs, one can draw inspiration from the numerous components based on MOFs with very large holes filled with different media proposed by Kerbage and Eggleton (hole assisted fibers) [EGG 01]. Indeed, the core of a PCF can be filled with different liquids (liquid crystal, amplifying medium, biological medium, liquid whose optical properties are modified by an outer event such as the application of an electric field or a change in temperature …). Obviously, even if the structure of the PCF is preserved, the operating guiding principle in this case is likely to become the TIR because of the higher index of medium in the core. Notwithstanding this, making the guided light wave interact with the liquid inserted into the core of the fiber opens the way to a multitude of applications: switches, tunable attenuators, tunable lazers, intrinsic sensors, biosensors for the analysis of sub-microliter samples, etc. [LAR 03].

3.7. Conclusion Since their quasi-fortuitous invention in 1996, interest in microstructured fibers guiding light following the total internal reflection principle has increased considerably. Such infatuation is due to the novel propagation properties of these fibers that open the way to a large number of applications not realizable with more conventional fibers, in particular for telecommunications and non-linear optics. Furthermore, the manufacturing technique of preforms that consist of stacking simple silica tubes and rods makes it possible to conceive structures with optogeometrical characteristics much more varied than those made possible through CVD-type techniques.

Photonic Crystal Fibers

129

Photonic crystal fibers operating by virtue of transverse resonance in cladding have also raised many hopes, especially since their ability to guide light in a hollow core over long distances has been experimentally demonstrated. This has been made possible by a spectacular improvement in fabrication techniques over the first years of the 21st century. The very first applications reported at the end of this chapter take advantage of the fact that, contrary to the case of all other fibers, the light wave can remain confined and propagate in low index media such as a vacuum, gas or certain liquids. The very long lengths of interaction between light and the medium inserted into the core enable us to efficiently exploit the optical properties of this medium: transparency, high or low non-linear coefficients, etc. With TIR-MOFs and PCFs, it has thus been shown that nanostructures could be realized and maintained longitudinally invariant in fibers over several hundreds of meters, even kilometers. Moreover, in another register, optical fibers can also form the basic element of short length nanostructures used in particular for near field analysis techniques. Indeed, the different techniques of near field microscopy are based on the use of probes consisting of a short sample of tapered fiber for making a sharp point (metallized or not) whose end is 50 to 200nm in diameter. In this case too, control of the geometry of the structure (shape of the tip, dimension of the nanoopening) is particularly critical. This issue is treated in detail in the Chapter 7. Considering the great efforts that the large laboratories have made to improve the performance of micro- and nanostructured fibers, one can predict that significant progress will be made shortly, in the early years of the twenty-first century, in particular in the fields of telecommunications, metrology and opto-biology. Even though the results already obtained are numerous and convincing, they are perhaps only the beginnings of a true revolution in the field of guided optics.

3.8. References [AUG 02] AUGUSTE J.L., BLONDY J.M., MAURY J., MARCOU J., DUSSARDIER B., MONNOM G., JINDAL R., THYAGARAJAN K., PAL B., “Conception, realization, and characterization of a very high negative chromatic dispersion fiber”, Optical Fiber Technology, 8, p. 89-105, 2002. [BAR 99] BARKOU S.E., BROENG J., BJARKLEV A., “Silica-air photonic crystal fiber design that permits waveguiding by true photonic bandgap effect”, Optics Letters, 24, p. 46-48, 1999. [BEN 02a] BENABID F., KNIGHT J.C., ANTONOPOULOS G., RUSSELL P.ST.J., “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber”, Science, 298, p. 399-402, 2002. [BEN 02b] BENABID F., KNIGHT J.C., RUSSELL P.ST.J., “Particle levitation and guidance in hollow-core photonic crystal fiber”, Optics Express, 10, p. 1195-1203, 2002.

130

Nanophotonics

[BER 94] BERENGER J.P., “A perfectly matched layer for the absorption of electromagnetic waves”, Journal of Computational Physics, 114, p. 185-200, 1994. [BIR 95] BIRKS T.A., ROBERTS P.J., RUSSELL P.ST.J., ATKIN D.M., SHEPHERD T.J., “Full 2-D photonic bandgaps in silica/air structures”, Electronics Letters, 31, p. 19411943, 1995. [BIR 97] BIRKS T.A., KNIGHT J.C., RUSSELL P.ST.J., “Endlessly single-mode photonic crystal fiber”, Optics Letters, 22, p. 961-963, 1997. [BIR 04] BIRKS T.A., BIRD D.M., HEDLEY T.D., POTTAGE J.M., RUSSELL P.ST.J., “Scaling laws and vector effects in bandgap-guiding fibres”, Optics Express, 12, p. 69-73, 2004. [BRE 00a] BRÉCHET F., LEPROUX P., ROY P., MARCOU J., PAGNOUX D., “Analysis of bandpass filtering behaviour of singlemode depressed-core-index photonic – bandgap fibre”, Electronics Letters, 36, p. 870-872, 2000. [BRE 00b] BRÉCHET F., MARCOU J., PAGNOUX D., ROY P., “Complete analysis of the propagation characteristics into photonic crystal fibers by the finite element method”, Optical Fiber Technology, 6, p. 181-191, 2000. [BRE 00c] BRÉCHET F., ROY P., MARCOU J., PAGNOUX D., “Single-mode propagation into depressed-core photonic-bandgap fibre designed for zero-dispersion propagation at short wavelengths”, Electronics Letters, 36, p. 514-515, 2000. [BRO 00] BROENG J., BARKOU S.E., SONDERGAARD T., BJARKLEV A., “Analysis of air-guiding photonic bandgap fibers”, Optics Letters, 25, p. 96-98, 2000. [CHA 04] CHAMPERT P.A., COUDERC V., LEPROUX P., FÉVRIER S., TOMBELAINE V., LABONTÉ L., ROY P., FROEHLY C., “White-light supercontinuum generation in normally dispersive optical fiber using original multi-wavelength pumping system”, Optics Express, 19, p. 4366-4371, 2004. [CRE 99] CREGAN R.F., MANGAN B.J., KNIGHT J.C., BIRKS T.A., RUSSELL P.ST.J., ROBERTS P.J., ALLAN D.C., “Single mode photonic band gap guidance of light in air”, Science, 285, p. 1537-1539, 1999. [CRI 04] CRISTIANI I., TEDIOSI R., TARTARA L., DEGIORGIO V., “Dispersive wave generation by solitons in microstructured optical fibers”, Optics Express, 12, p. 124-135, 2004. [DIG 04] DIGONNET M.J.F., KIM H.K., SHIN J., FAN S., KINO G.S., “Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers”, Optics Express, 9, p. 1864-1872, 2004. [EGG 01] EGGLETON B.J., KERBAGE C., WESTBROOK P.S., WINDELER R.S., HALE A., “Microstructured optical devices”, Optics Express, 9, p. 698-713, 2001. [FER 99] FERRANDO A., SILVESTRE F., MIRET J.J., ANDRÈS P., ANDRÈS M.V., “Full-vector analysis of a realistic photonic crystal fiber”, Optics Letters, 24, p. 276-278, 1999.

Photonic Crystal Fibers

131

[FER 00] FERRANDO A., SILVESTRE E., MIRET J.J., ANDRÈS M.V., “Nearly zero ultraflattened dispersion in photonic crystal fibers”, Optics Letters, 25, p. 790-792, 2000. [FEV 01] FÉVRIER S., ALBERT A., LOURADOUR F., ROY P., PAGNOUX D., BARTHÉLÉMY A., “Fibres optiques non linéaires microstructurées pour source lumineuse blanche d’impulsions femtosecondes”, Proceedings of COLOQ 2001, Rennes, 2001. [FIN 99] FINK Y., RIPIN J., SHANHUI F., CHIPING C., JOANNOPOULOS J.D., THOMAS E.L., “Guiding optical light in air using an all-dielectric structure”, Journal of Lightwave Technology, 17, p. 2039-2041, 1999. [FOG 02] FOGLI F., SACCOMANDI L., BASSI P., BELLANCA G., TRILLO S., “Full vectorial BPM modeling in index-guiding photonic crystal fibers and couplers”, Optics Express, 10, p. 54-59, 2002. [HAN 03] HANSEN T.P., BROENG J., JAKOBSEN C., VIENNE G., SIMONSEN H.R., NIELSEN M.D., SKOVAGAARD P.M.W., FOLKENBERG J.R., BJARKLEV A., “Air guidance over 345m of large-core photonic bandgap fiber”, Proceedings of Optical Fiber Conference, paper PD4, Atlanta, 2003. [HIL 03] HILAIRE S., ROY P., PAGNOUX D., BAYART D., “Large mode Er3+ – doped photonic crystal fibre amplifier for higly efficient amplification”, Proceedings of European Conference on Optical Communications, paper We4.P13, Rimini, 2003. [HOL 00] HOLSWARTH R., UDEM T., HÄNSCH T.W., KNIGHT J.C., WADSWORTH W.J., RUSSELL P.ST.J., “Optical frequency synthesizer for precision spectroscopy”, Physical Review Letters, 85, p. 2264-2266, 2000. [KIM 00] KIM J., CHUNG Y., PAEK U.C., KIM D.Y., “A new numerical design tool for holey optical fibers”, Optoelectronics and communications conference, paper 12B3-3, Chiba Japon, 2000. [KNI 96] KNIGHT J.C., BIRKS T.A., RUSSELL P.ST.J., ATKIN D.M., “All-silica singlemode fiber with photonic crystal cladding”, Optics Letters, 21, p. 1547-1549, 1996. [KNI 98a] KNIGHT J.C., BIRKS T.A., CREGAN R.F., RUSSELL P.ST.J., DE SANDRO J.P., “Large mode area photonic crystal fibre”, Electronics Letters, 34, p. 1347-1348, 1998. [KNI 98b] KNIGHT J.C., BROENG J., BIRKS T.A., RUSSELL P.ST.J., “Photonic band gap guidance in optical fibers”, Science, 282, p. 1476-1478, 1998. [KNI 00] KNIGHT J.C., ARRIAGA J., BIRKS T.A., ORTIGOSA-BLANCH A., WADSWORTH W.J., RUSSELL P.ST.J., “Anomalous dispersion in photonic crystal fibres”, IEEE Photonics Technology Letters, 12, p. 807-809, 2000. [KUH 02] KUHLMEY B., WHITE T.P., RENVERSEZ G., MAYSTRE D., BOTTEN L.C., MARTIJN DE STERKE C., MCPHEDRAN R.C., “Multipole method for microstructured optical fibers II: implementation and results”, Journal of Optical Society of America, B 10, p. 2331-2340, 2002. [KUR 04] KURIKI K., SHAPIRA O., HART S.D., BENOIT G., KURIKI Y., VIENS J.F., BAYINDIR M., JOANNOPOULOS J.D., KINK Y., “Hollow multilayer photonic bandgap fibers for NIR applications”, Optics Express, 8, p. 1510-1517, 2004.

132

Nanophotonics

[LAR 03] LARSEN T.T., BJARKLEV A., HERMANN D.S., BROENG J., “Optical devices based on liquid crystal photonic bandgap fibres”, Optics Express, 11, p. 2589-2596, 2003. [LIB 01] LIBORI S.B., KNUDSEN E., BJARKLEV A., SIMONSEN H.R., “Highbirefringent photonic crystal fiber”, Proceedings of Optical Fiber Conference, paper TuM2, Anaheim, 2001. [LOU 05] LOURTIOZ J.M., BENISTY H., BERGER V., GÉRARD J.M., MAYSTRE D. and TCHELNOKOV A. (2005) Photonic Crystals: Towards Nanoscale Photonic Devices, Ed. Springer, Berlin, Heidelberg, New York. [MAN 04] MANGAN B.J., FARR L., LANGFORD A., ROBERTS P.J., WILLIAMS D.P., COUNY F., LAWMAN M., MASON M., COUPLAND S., FLEA R., SABERT H., “Low loss (1.7dB/km) hollow core photonic bandgap fiber”, Proceedings of Optical Fiber Conference, post-dead line paper PDP24, Anaheim, 2004. [MCN 03] MCNAB S.J., MOLL N., VLASOV Y.A., “Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides”, Optics Express, 22, p. 29272939, 2003. [MEL 03] MÉLIN G., CAVANI O., GASCA L., PEYRILLOUX A., PROVOST L., RÉJEAUNIER X., “Characterization of a polarization maintaining fibre”, Proceedings of European Conference on Optical Communications, paper We1.7.4, Rimini, 2003. [MEU 03] MEUNIER J.P., Physique et technologie des fibres optiques, Chapter 5, Hermès, Paris, 2003. [MON 99] MONRO T.M., RICHARDSON D.J., BRODERICK N.G.R., BENNETT P.J., “Holey optical fibers: an efficient modal model”, Journal of Lightwave Technology, 17, p. 1093-1102, 1999. [MON 00] MONRO T.M., RICHARDSON D.J., BRODERICK N.G.R., BENNETT P.J., “Modeling large air fraction holey optical fibers”, Journal of Lightwave Technology, 18, p. 50-56, 2000. [NED 86] NEDELEC J.C., “A new family of mixed finite element in R3”, Numer. Math., 50, p. 57-81, 1986. [NIE 02] NIEMI T., LUDVIGSEN H., SCHOLDER F., LEGRÉ M., WEGMULLER M., GISIN N., JENSEN J.R., PETERSSON A., SKOVGAARD P.M.W., “Polarization properties of single-moded, large-mode area photonic crystal fibers”, Proceedings of European Conference on Optical Communications, paper 1.9, Copenhague, 2002. [ORT 00] ORTIGOSA-BLANCH A., KNIGHT J.C., WADSWORTH W.J., ARRIAGA J., MANGAN B.J., BIRKS T.A., RUSSELL P.ST.J., “Highly birefringent photonic crystal fibers”, Optics Letters, 25, p. 1325-1327, 2000. [OUZ 03] OUZOUNOV D.G., AHMAD F.R., MÜLLER D., VENKATARAMAN N., GALLAGHER M.T., THOMAS M.G., SILCOX J., KOCH K.W., GAETA A.L., “Generation of megawatt optical solitons in hollow-core photonic crystal band-gap fibers”, Science, 301, p. 1702-1704, 2003. [PEN 94] PENDRY J.B., “Photonic band structures”, Journal of Modern Optics, 41, p. 209229, 1994.

Photonic Crystal Fibers

133

[PEY 02a] PEYRILLOUX A., FÉVRIER S., MARCOU J., BERTHELOT L., PAGNOUX D., SANSONETTI P., “Comparison between the finite element method, the localized function method and a novel equivalent averaged index method for modelling photonic crystal fibres”, Journal of Optics A: Pure and Applied Optics, 4, p. 257-262, 2002. [PEY 02b] PEYRILLOUX A., PAGNOUX D., REYNAUD F., “Evaluation of photonic crystal fiber potential for fiber linked version of stellar interferometers”, Proceedings of SPIE conference “Astronomical Telescopes and Instrumentation”, Hawaii, 2002. [PEY 03a] PEYRILLOUX A., Modélisation et caractérisation des fibres microstructurées air/silice pour applications aux télécommunications optiques, Thesis, University of Limoges, 2003. [PEY 03b] PEYRILLOUX A., CHARTIER T., HIDEUR A., BERTHELOT L., MELIN G., GASCA L., PAGNOUX D., ROY P., “Theoretical and experimental study of the birefringence of a photonic crystal fiber”, Journal of Lightwave Technology 21, p. 536539, 2003. [POT 03] POTTAGE J.M., BIRD D.M., HEDLEY T.D., BIRKS T.A., KNIGHT J.C., RUSSELL P.ST.J., “Robust photonic band gaps for hollow core guidance in PCF made from high index glass”, Optics Express, 11, p. 2854-2861, 2003. [PRO 01] PROVINO L., DUDLEY J.M., MAILLOTTE H., GROSSARD N., WINDELER R.S., EGGLETON B.J., “Compact broadband continuum source based on microchip laser pumped microstructured fibre”, Electronics Letters, 37, p. 558-560, 2001. [RAN 00] RANKA J.K., WINDELER R.S., STENTZ A.J., “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800nm”, Optics Letters, 25, p. 25-27, 2000. [RAS 03] RASTOGI V., CHIANG K.S., “Holey optical fiber with circulary distributed holes analysed by the radial effective-index method”, Optics Letters, 28, p. 2449-2451, 2003. [REE 02] REEVES W.H., KNIGHT J.C., RUSSELL P.ST.J., ROBERTS P.J., “Demonstration of ultra-flattened dispersion in photonic crystal fibers”, Optics Express, 10, p. 609-613, 2002. [RUS 01] RUSSELL P.S.J., “Holey new fibers”, Tutorial in Optical Fiber Conference, TuL1, Anaheim, 2001. [SAI 03] SAITOH K., KOSHIBA M., “Leakage loss and group velocity dispersion in air-core photonic bandgap fibers”, Optics Express, 11, p. 3100-3109, 2003. [SHA 02a] SHARPING J.E., FIORENTINO M., KUMAR P., WINDELER R.S., “All-optical switching based on cross phase modulation in microstructure fiber”, IEEE Photonics Technology Letters, 14, p. 77-79, 2002. [SHA 02b] SHARPING J.E., FIORENTINO M., KUMAR P., WINDELER R.S., “Opticalparametric oscillator based on four-wave mixing in microstructure fiber”, Optics Letters, 27, p. 1675-1677, 2002. [SIL 98] SILVESTRE F., ANDRÈS M.V., ANDRÈS P., “Biorthonormal-basis method for the vector description of optical-fibre modes”, Journal of Lightwave Technology 16, p. 923928, 1998.

134

Nanophotonics

[SOR 01] SORENSEN T., BROENG J., BJARKLEV A., KNUDSEN E., BARKOU LIBORI S.E., “Macro-bending loss of photonic crystal fibre”, Electronics Letters, 37, p. 287-289, 2001. [STE 01] STEEL M.J., WHITE T.P., MARTIJN DE STERKE C., MCPHEDRAN R.C., BOTTEN L.C., “Symmetry and degeneracy in microstructured optical fibers”, Optics Letters, 26, p. 488-490, 2001. [TAJ 03] TAJIMA K., ZHOU J., NAKAJIMA K., SATO K., “Low water peak photonic crystal fiber”, Proceedings of European Conference on Optical Communications, paper PD34, Rimini, 2003. [TAI 03] TAILLAERT D., CHONG H., BOREL P.I., FRANDSEN L.H., DE LA RUE R., BAETS R., “A compact two-dimmensional grating coupler used as a polarization splitter”, IEEE Photonics Technology Letters, 9, p. 1249-1251, 2003. [VEN 02] VENKATARAMAN N., GALLAGHER M.T., SMITH C.M., MÜLLER D., WEST J.A., KOCH K.W., FAJARDO J.C., “Low loss (13dB/km) air core photonic bandgap fibre”, Proceedings of European Conference on Optical Communications, paper PD1.1, Copenhagen, 2002. [VIL 92] VILLENEUVE P.R., PICHÉ M., “Photonic band gaps in two-dimensional square and hexaganal lattices”, Physical Review, B 46, p. 4969-4972, 1992. [WAD 04] WADSWORTH W.J., JOLY N., KNIGHT J.C., BIRKS T.A., BIANCALANA F., RUSSELL P.ST.J., “Supercontinuum and four-wave mixing with Q-switched pulses in endlessly single-mode photonic crystal fibres”, Optics Express, 12, p. 299-309, 2004. [WES 00] WEST J.A., FAJARDO J.C., GALLAGHER M.T., KOCH K.W., BORRELLI N.F., ALLAN D.C., “Demonstration of IR-optimized air-core photonic band-gap fiber”, Proceedings of European Conference on Optical Communications, p. 41-42, Munich, 2000. [WES 01] WEST J.A., VENKATARAMAN N., SMITH C.M., GALLAGHER M.T., “Photonic crystal fibres”, Proceedings of European Conference on Optical Communications, paper Th A22, Amsterdam, 2001. [WHI 01] WHITE T.P., MCPHEDRAN R.C., DE STERKE C.M., “Confinement losses in microstructured optical fibers”, Optics Letters, 26, p. 1660-1662, 2001. [WHI 02] WHITE T.P., KUHLMEY B., MCPHEDRAN R.C., MAYSTRE D., RENVERSEZ G., MARTIJN DE STERKE C., BOTTEN L.C., “Multipole method for microstructured optical fibers I: formulation”, Journal of Optical Society of America, B 10, p. 2322-2330, 2002. [YEH 78] YEH P., YARIV A., MAROM E., “Theory of Bragg fiber”, Journal of Optical Society of America, 68, p. 1196-1201, 1978. [YU 04] YU C.P., CHANG H.C., “Yee-mesh based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers”, Optics Express, 12, p. 6165-6177, 2004.

Chapter 4

Quantum Dots in Optical Microcavities

4.1. Introduction Since 1990, Cavity Quantum ElectroDynamics (CQED) has become a major source of inspiration for basic research in optoelectronics [BUR 95; WEI 96; DUC 96; BEN 98]. In the 1980s, a beautiful series of experiments on atoms in microwave and optical cavities had demonstrated that optical processes – including spontaneous emission (SE) – can be deeply modified using a cavity to tailor the emitter-field coupling [HAR 89]. Among other effects observable in the so called “weak coupling regime”, the modification of the emission diagram, the enhancement or inhibition of the SE rate, the funneling of SE photons into a single mode and the control of the SE process on the single photon level are particularly attractive for applications in optoelectronics. For very high-Q (i.e. weakly damped) cavities, SE can even become a reversible process, in the so called “strong-coupling regime”. Improving control over spontaneous emission processes in optoelectronic devices was recognized in the mid-1980s as a promising avenue for improving the performance of light emitting diodes and lazer diodes. In lazers, for instance, photons that are spontaneously emitted into the lasing mode(s) act as seeds for amplification by stimulated emission. Conversely, SE into other modes is useless and even detrimental, as it consumes a significant amount of the injected electronhole pairs.

Chapter written by Jean-Michel GÉRARD.

136

Nanophotonics

By increasing the fraction β of the SE that is coupled to the lasing mode, one can reduce (roughly in inverse proportion) the threshold current of the lazer. Novel devices based on full spontaneous emission control have also been proposed. In the β=1 limit, all photons (be they emitted by SE or by stimulated emission) are funneled into the lasing mode. This device, known as “thresholdless lazer” [YOK 92; BJO 91], would act as an ideal converter of electrical signals into optical signals. Another attractive device is the single mode single photon source, which is able to generate on demand light pulses containing a single photon prepared in a well defined quantum state (spatial mode, polarization …). Such a source can be obtained by placing a single quantum emitter (atom, ion, quantum dot …), able to generate photons one by one, into a single mode optical microcavity. High quality solid-state microcavities became widely available in the early 1990s, and have since brought about major achievements, such as the observation of strong coupling for quantum wells in planar cavities [WEI 92], the fabrication of high-efficiency microcavity LEDs which exploit SE angular redistribution [DEN 97], and low-threshold vertical-cavity surface emitting lazers [HUF 97] or microsphere lazers [SAN 96]. However, the emission spectrum of solid-state emitters is usually spectrally broad, which has long been a major hindrance to the observation of several important CQED effects. Because of their “atomic-like” properties, self-assembled semiconductor quantum dots (QDs) are particularly well suited to performing solid-state CQED experiments. By placing one or a few QDs in semiconductor optical microcavities, one can for instance obtain the first clear demonstrations, in a solid-state system, of the Purcell effect [GER 98; GRA 99; GAY 98; GAY 01; KIR 01; MOR 01a; SOL 01; VUC 03] and of the strong coupling regime for a single discrete emitter [REI 04; YOS 04; PET 05]. A single-mode solid-state source of single photons based on a single QD in a pillar microcavity has also been developed [MOR 01a; SAN 02; OLI 05]. This device is the first optoelectronic component whose operation actually relies on CQED, through the Purcell effect, and has a strong application potential in the field of quantum communications and quantum information processing. In this chapter, we first briefly present in section 4.2 some relevant properties of QDs as well as available single mode semiconductor microcavities. Basic CQED effects observed in the weak and strong coupling regimes are discussed in section 4.3. Finally, well-established as well as more prospective applications of these CQED effects in the field of nano-optoelectronics and quantum information processing are finally discussed in section 4.4.

Quantum Dots in Optical Microcavities

137

4.2. Building blocks for solid-state CQED 4.2.1. Self-assembled QDs as “artificial atoms” Solid-state CQED experiments have until now generally been performed on InAs/GaAs QDs in GaAs/GaAlAs microcavities. We will concentrate our attention on this very mature system, although other QDs, such as II–VI self-assembled QDs [ROB 05], QDs formed by interface fluctuations in quantum wells [BRU 94; HOU 03; PET 05] and semiconductor nanocrystals [FAN 00; ART 01] are also potentially interesting in this context. It has been known since 1985 [GOL 85] that strained-layer epitaxy can be used to build defect-free nanometer scale InAs rich clusters in GaAs, which constitute potential traps for both electrons and holes. These self-assembled QDs support wellseparated discrete electronic states and exhibit a single narrow emission line (